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A fundamental and recurring problem in analytic number theory is to demonstrate the presence of cancellation in an oscillating sum, a typical example of which might be a correlation

\displaystyle  \sum_{n} f(n) \overline{g(n)} \ \ \ \ \ (1)

between two arithmetic functions {f: {\bf N} \rightarrow {\bf C}} and {g: {\bf N} \rightarrow {\bf C}}, which to avoid technicalities we will assume to be finitely supported (or that the {n} variable is localised to a finite range, such as {\{ n: n \leq x \}}). A key example to keep in mind for the purposes of this set of notes is the twisted von Mangoldt summatory function

\displaystyle  \sum_{n \leq x} \Lambda(n) \overline{\chi(n)} \ \ \ \ \ (2)

that measures the correlation between the primes and a Dirichlet character {\chi}. One can get a “trivial” bound on such sums from the triangle inequality

\displaystyle  |\sum_{n} f(n) \overline{g(n)}| \leq \sum_{n} |f(n)| |g(n)|;

for instance, from the triangle inequality and the prime number theorem we have

\displaystyle  |\sum_{n \leq x} \Lambda(n) \overline{\chi(n)}| \leq x + o(x) \ \ \ \ \ (3)

as {x \rightarrow \infty}. But the triangle inequality is insensitive to the phase oscillations of the summands, and often we expect (e.g. from the probabilistic heuristics from Supplement 4) to be able to improve upon the trivial triangle inequality bound by a substantial amount; in the best case scenario, one typically expects a “square root cancellation” that gains a factor that is roughly the square root of the number of summands. (For instance, for Dirichlet characters {\chi} of conductor {O(x^{O(1)})}, it is expected from probabilistic heuristics that the left-hand side of (3) should in fact be {O_\varepsilon(x^{1/2+\varepsilon})} for any {\varepsilon>0}.)

It has proven surprisingly difficult, however, to establish significant cancellation in many of the sums of interest in analytic number theory, particularly if the sums do not have a strong amount of algebraic structure (e.g. multiplicative structure) which allow for the deployment of specialised techniques (such as multiplicative number theory techniques). In fact, we are forced to rely (to an embarrassingly large extent) on (many variations of) a single basic tool to capture at least some cancellation, namely the Cauchy-Schwarz inequality. In fact, in many cases the classical case

\displaystyle  |\sum_n f(n) \overline{g(n)}| \leq (\sum_n |f(n)|^2)^{1/2} (\sum_n |g(n)|^2)^{1/2}, \ \ \ \ \ (4)

considered by Cauchy, where at least one of {f, g: {\bf N} \rightarrow {\bf C}} is finitely supported, suffices for applications. Roughly speaking, the Cauchy-Schwarz inequality replaces the task of estimating a cross-correlation between two different functions {f,g}, to that of measuring self-correlations between {f} and itself, or {g} and itself, which are usually easier to compute (albeit at the cost of capturing less cancellation). Note that the Cauchy-Schwarz inequality requires almost no hypotheses on the functions {f} or {g}, making it a very widely applicable tool.

There is however some skill required to decide exactly how to deploy the Cauchy-Schwarz inequality (and in particular, how to select {f} and {g}); if applied blindly, one loses all cancellation and can even end up with a worse estimate than the trivial bound. For instance, if one tries to bound (2) directly by applying Cauchy-Schwarz with the functions {\Lambda} and {\chi}, one obtains the bound

\displaystyle  |\sum_{n \leq x} \Lambda(n) \overline{\chi(n)}| \leq (\sum_{n \leq x} \Lambda(n)^2)^{1/2} (\sum_{n \leq x} |\chi(n)|^2)^{1/2}.

The right-hand side may be bounded by {\ll x \log^{1/2} x}, but this is worse than the trivial bound (3) by a logarithmic factor. This can be “blamed” on the fact that {\Lambda} and {\chi} are concentrated on rather different sets ({\Lambda} is concentrated on primes, while {\chi} is more or less uniformly distributed amongst the natural numbers); but even if one corrects for this (e.g. by weighting Cauchy-Schwarz with some suitable “sieve weight” that is more concentrated on primes), one still does not do any better than (3). Indeed, the Cauchy-Schwarz inequality suffers from the same key weakness as the triangle inequality: it is insensitive to the phase oscillation of the factors {f, g}.

While the Cauchy-Schwarz inequality can be poor at estimating a single correlation such as (1), its power improves when considering an average (or sum, or square sum) of multiple correlations. In this set of notes, we will focus on one such situation of this type, namely that of trying to estimate a square sum

\displaystyle  (\sum_{j=1}^J |\sum_{n} f(n) \overline{g_j(n)}|^2)^{1/2} \ \ \ \ \ (5)

that measures the correlations of a single function {f: {\bf N} \rightarrow {\bf C}} with multiple other functions {g_j: {\bf N} \rightarrow {\bf C}}. One should think of the situation in which {f} is a “complicated” function, such as the von Mangoldt function {\Lambda}, but the {g_j} are relatively “simple” functions, such as Dirichlet characters. In the case when the {g_j} are orthonormal functions, we of course have the classical Bessel inequality:

Lemma 1 (Bessel inequality) Let {g_1,\dots,g_J: {\bf N} \rightarrow {\bf C}} be finitely supported functions obeying the orthonormality relationship

\displaystyle  \sum_n g_j(n) \overline{g_{j'}(n)} = 1_{j=j'}

for all {1 \leq j,j' \leq J}. Then for any function {f: {\bf N} \rightarrow {\bf C}}, we have

\displaystyle  (\sum_{j=1}^J |\sum_{n} f(n) \overline{g_j(n)}|^2)^{1/2} \leq (\sum_n |f(n)|^2)^{1/2}.

For sake of comparison, if one were to apply the Cauchy-Schwarz inequality (4) separately to each summand in (5), one would obtain the bound of {J^{1/2} (\sum_n |f(n)|^2)^{1/2}}, which is significantly inferior to the Bessel bound when {J} is large. Geometrically, what is going on is this: the Cauchy-Schwarz inequality (4) is only close to sharp when {f} and {g} are close to parallel in the Hilbert space {\ell^2({\bf N})}. But if {g_1,\dots,g_J} are orthonormal, then it is not possible for any other vector {f} to be simultaneously close to parallel to too many of these orthonormal vectors, and so the inner products of {f} with most of the {g_j} should be small. (See this previous blog post for more discussion of this principle.) One can view the Bessel inequality as formalising a repulsion principle: if {f} correlates too much with some of the {g_j}, then it does not have enough “energy” to have large correlation with the rest of the {g_j}.

In analytic number theory applications, it is useful to generalise the Bessel inequality to the situation in which the {g_j} are not necessarily orthonormal. This can be accomplished via the Cauchy-Schwarz inequality:

Proposition 2 (Generalised Bessel inequality) Let {g_1,\dots,g_J: {\bf N} \rightarrow {\bf C}} be finitely supported functions, and let {\nu: {\bf N} \rightarrow {\bf R}^+} be a non-negative function. Let {f: {\bf N} \rightarrow {\bf C}} be such that {f} vanishes whenever {\nu} vanishes, we have

\displaystyle  (\sum_{j=1}^J |\sum_{n} f(n) \overline{g_j(n)}|^2)^{1/2} \leq (\sum_n |f(n)|^2 / \nu(n))^{1/2} \ \ \ \ \ (6)

\displaystyle  \times ( \sum_{j=1}^J \sum_{j'=1}^J c_j \overline{c_{j'}} \sum_n \nu(n) g_j(n) \overline{g_{j'}(n)} )^{1/2}

for some sequence {c_1,\dots,c_J} of complex numbers with {\sum_{j=1}^J |c_j|^2 = 1}, with the convention that {|f(n)|^2/\nu(n)} vanishes whenever {f(n), \nu(n)} both vanish.

Note by relabeling that we may replace the domain {{\bf N}} here by any other at most countable set, such as the integers {{\bf Z}}. (Indeed, one can give an analogue of this lemma on arbitrary measure spaces, but we will not do so here.) This result first appears in this paper of Boas.

Proof: We use the method of duality to replace the role of the function {f} by a dual sequence {c_1,\dots,c_J}. By the converse to Cauchy-Schwarz, we may write the left-hand side of (6) as

\displaystyle  \sum_{j=1}^J \overline{c_j} \sum_{n} f(n) \overline{g_j(n)}

for some complex numbers {c_1,\dots,c_J} with {\sum_{j=1}^J |c_j|^2 = 1}. Indeed, if all of the {\sum_{n} f(n) \overline{g_j(n)}} vanish, we can set the {c_j} arbitrarily, otherwise we set {(c_1,\dots,c_J)} to be the unit vector formed by dividing {(\sum_{n} f(n) \overline{g_j(n)})_{j=1}^J} by its length. We can then rearrange this expression as

\displaystyle  \sum_n f(n) \overline{\sum_{j=1}^J c_j g_j(n)}.

Applying Cauchy-Schwarz (dividing the first factor by {\nu(n)^{1/2}} and multiplying the second by {\nu(n)^{1/2}}, after first removing those {n} for which {\nu(n)} vanish), this is bounded by

\displaystyle  (\sum_n |f(n)|^2 / \nu(n))^{1/2} (\sum_n \nu(n) |\sum_{j=1}^J c_j g_j(n)|^2)^{1/2},

and the claim follows by expanding out the second factor. \Box

Observe that Lemma 1 is a special case of Proposition 2 when {\nu=1} and the {g_j} are orthonormal. In general, one can expect Proposition 2 to be useful when the {g_j} are almost orthogonal relative to {\nu}, in that the correlations {\sum_n \nu(n) g_j(n) \overline{g_{j'}(n)}} tend to be small when {j,j'} are distinct. In that case, one can hope for the diagonal term {j=j'} in the right-hand side of (6) to dominate, in which case one can obtain estimates of comparable strength to the classical Bessel inequality. The flexibility to choose different weights {\nu} in the above proposition has some technical advantages; for instance, if {f} is concentrated in a sparse set (such as the primes), it is sometimes useful to tailor {\nu} to a comparable set (e.g. the almost primes) in order not to lose too much in the first factor {\sum_n |f(n)|^2 / \nu(n)}. Also, it can be useful to choose a fairly “smooth” weight {\nu}, in order to make the weighted correlations {\sum_n \nu(n) g_j(n) \overline{g_{j'}(n)}} small.

Remark 3 In harmonic analysis, the use of tools such as Proposition 2 is known as the method of almost orthogonality, or the {TT^*} method. The explanation for the latter name is as follows. For sake of exposition, suppose that {\nu} is never zero (or we remove all {n} from the domain for which {\nu(n)} vanishes). Given a family of finitely supported functions {g_1,\dots,g_J: {\bf N} \rightarrow {\bf C}}, consider the linear operator {T: \ell^2(\nu^{-1}) \rightarrow \ell^2(\{1,\dots,J\})} defined by the formula

\displaystyle  T f := ( \sum_{n} f(n) \overline{g_j(n)} )_{j=1}^J.

This is a bounded linear operator, and the left-hand side of (6) is nothing other than the {\ell^2(\{1,\dots,J\})} norm of {Tf}. Without any further information on the function {f} other than its {\ell^2(\nu^{-1})} norm {(\sum_n |f(n)|^2 / \nu(n))^{1/2}}, the best estimate one can obtain on (6) here is clearly

\displaystyle  (\sum_n |f(n)|^2 / \nu(n))^{1/2} \times \|T\|_{op},

where {\|T\|_{op}} denotes the operator norm of {T}.

The adjoint {T^*: \ell^2(\{1,\dots,J\}) \rightarrow \ell^2(\nu^{-1})} is easily computed to be

\displaystyle  T^* (c_j)_{j=1}^J := (\sum_{j=1}^J c_j \nu(n) g_j(n) )_{n \in {\bf N}}.

The composition {TT^*: \ell^2(\{1,\dots,J\}) \rightarrow \ell^2(\{1,\dots,J\})} of {T} and its adjoint is then given by

\displaystyle  TT^* (c_j)_{j=1}^J := (\sum_{j=1}^J c_j \sum_n \nu(n) g_j(n) \overline{g_{j'}}(n) )_{j=1}^J.

From the spectral theorem (or singular value decomposition), one sees that the operator norms of {T} and {TT^*} are related by the identity

\displaystyle  \|T\|_{op} = \|TT^*\|_{op}^{1/2},

and as {TT^*} is a self-adjoint, positive semi-definite operator, the operator norm {\|TT^*\|_{op}} is also the supremum of the quantity

\displaystyle  \langle TT^* (c_j)_{j=1}^J, (c_j)_{j=1}^J \rangle_{\ell^2(\{1,\dots,J\})} = \sum_{j=1}^J \sum_{j'=1}^J c_j \overline{c_{j'}} \sum_n \nu(n) g_j(n) \overline{g_{j'}(n)}

where {(c_j)_{j=1}^J} ranges over unit vectors in {\ell^2(\{1,\dots,J\})}. Putting these facts together, we obtain Proposition 2; furthermore, we see from this analysis that the bound here is essentially optimal if the only information one is allowed to use about {f} is its {\ell^2(\nu^{-1})} norm.

For further discussion of almost orthogonality methods from a harmonic analysis perspective, see Chapter VII of this text of Stein.

Exercise 4 Under the same hypotheses as Proposition 2, show that

\displaystyle  \sum_{j=1}^J |\sum_{n} f(n) \overline{g_j(n)}| \leq (\sum_n |f(n)|^2 / \nu(n))^{1/2}

\displaystyle  \times ( \sum_{j=1}^J \sum_{j'=1}^J |\sum_n \nu(n) g_j(n) \overline{g_{j'}(n)}| )^{1/2}

as well as the variant inequality

\displaystyle  |\sum_{j=1}^J \sum_{n} f(n) \overline{g_j(n)}| \leq (\sum_n |f(n)|^2 / \nu(n))^{1/2}

\displaystyle  \times | \sum_{j=1}^J \sum_{j'=1}^J \sum_n \nu(n) g_j(n) \overline{g_{j'}(n)}|^{1/2}.

Proposition 2 has many applications in analytic number theory; for instance, we will use it in later notes to control the large value of Dirichlet series such as the Riemann zeta function. One of the key benefits is that it largely eliminates the need to consider further correlations of the function {f} (other than its self-correlation {\sum_n |f(n)|^2 / \nu(n)} relative to {\nu^{-1}}, which is usually fairly easy to compute or estimate as {\nu} is usually chosen to be relatively simple); this is particularly useful if {f} is a function which is significantly more complicated to analyse than the functions {g_j}. Of course, the tradeoff for this is that one now has to deal with the coefficients {c_j}, which if anything are even less understood than {f}, since literally the only thing we know about these coefficients is their square sum {\sum_{j=1}^J |c_j|^2}. However, as long as there is enough almost orthogonality between the {g_j}, one can estimate the {c_j} by fairly crude estimates (e.g. triangle inequality or Cauchy-Schwarz) and still get reasonably good estimates.

In this set of notes, we will use Proposition 2 to prove some versions of the large sieve inequality, which controls a square-sum of correlations

\displaystyle  \sum_n f(n) e( -\xi_j n )

of an arbitrary finitely supported function {f: {\bf Z} \rightarrow {\bf C}} with various additive characters {n \mapsto e( \xi_j n)} (where {e(x) := e^{2\pi i x}}), or alternatively a square-sum of correlations

\displaystyle  \sum_n f(n) \overline{\chi_j(n)}

of {f} with various primitive Dirichlet characters {\chi_j}; it turns out that one can prove a (slightly sub-optimal) version of this inequality quite quickly from Proposition 2 if one first prepares the sum by inserting a smooth cutoff with well-behaved Fourier transform. The large sieve inequality has many applications (as the name suggests, it has particular utility within sieve theory). For the purposes of this set of notes, though, the main application we will need it for is the Bombieri-Vinogradov theorem, which in a very rough sense gives a prime number theorem in arithmetic progressions, which, “on average”, is of strength comparable to the results provided by the Generalised Riemann Hypothesis (GRH), but has the great advantage of being unconditional (it does not require any unproven hypotheses such as GRH); it can be viewed as a significant extension of the Siegel-Walfisz theorem from Notes 2. As we shall see in later notes, the Bombieri-Vinogradov theorem is a very useful ingredient in sieve-theoretic problems involving the primes.

There is however one additional important trick, beyond the large sieve, which we will need in order to establish the Bombieri-Vinogradov theorem. As it turns out, after some basic manipulations (and the deployment of some multiplicative number theory, and specifically the Siegel-Walfisz theorem), the task of proving the Bombieri-Vinogradov theorem is reduced to that of getting a good estimate on sums that are roughly of the form

\displaystyle  \sum_{j=1}^J |\sum_n \Lambda(n) \overline{\chi_j}(n)| \ \ \ \ \ (7)

for some primitive Dirichlet characters {\chi_j}. This looks like the type of sum that can be controlled by the large sieve (or by Proposition 2), except that this is an ordinary sum rather than a square sum (i.e., an {\ell^1} norm rather than an {\ell^2} norm). One could of course try to control such a sum in terms of the associated square-sum through the Cauchy-Schwarz inequality, but this turns out to be very wasteful (it loses a factor of about {J^{1/2}}). Instead, one should try to exploit the special structure of the von Mangoldt function {\Lambda}, in particular the fact that it can be expressible as a Dirichlet convolution {\alpha * \beta} of two further arithmetic sequences {\alpha,\beta} (or as a finite linear combination of such Dirichlet convolutions). The reason for introducing this convolution structure is through the basic identity

\displaystyle  (\sum_n \alpha*\beta(n) \overline{\chi_j}(n)) = (\sum_n \alpha(n) \overline{\chi_j}(n)) (\sum_n \beta(n) \overline{\chi_j}(n)) \ \ \ \ \ (8)

for any finitely supported sequences {\alpha,\beta: {\bf N} \rightarrow {\bf C}}, as can be easily seen by multiplying everything out and using the completely multiplicative nature of {\chi_j}. (This is the multiplicative analogue of the well-known relationship {\widehat{f*g}(\xi) = \hat f(\xi) \hat g(\xi)} between ordinary convolution and Fourier coefficients.) This factorisation, together with yet another application of the Cauchy-Schwarz inequality, lets one control (7) by square-sums of the sort that can be handled by the large sieve inequality.

As we have seen in Notes 1, the von Mangoldt function {\Lambda} does indeed admit several factorisations into Dirichlet convolution type, such as the factorisation {\Lambda = \mu * L}. One can try directly inserting this factorisation into the above strategy; it almost works, however there turns out to be a problem when considering the contribution of the portion of {\mu} or {L} that is supported at very small natural numbers, as the large sieve loses any gain over the trivial bound in such settings. Because of this, there is a need for a more sophisticated decomposition of {\Lambda} into Dirichlet convolutions {\alpha * \beta} which are non-degenerate in the sense that {\alpha,\beta} are supported away from small values. (As a non-example, the trivial factorisation {\Lambda = \Lambda * \delta} would be a totally inappropriate factorisation for this purpose.) Fortunately, it turns out that through some elementary combinatorial manipulations, some satisfactory decompositions of this type are available, such as the Vaughan identity and the Heath-Brown identity. By using one of these identities we will be able to complete the proof of the Bombieri-Vinogradov theorem. (These identities are also useful for other applications in which one wishes to control correlations between the von Mangoldt function {\Lambda} and some other sequence; we will see some examples of this in later notes.)

For further reading on these topics, including a significantly larger number of examples of the large sieve inequality, see Chapters 7 and 17 of Iwaniec and Kowalski.

Remark 5 We caution that the presentation given in this set of notes is highly ahistorical; we are using modern streamlined proofs of results that were first obtained by more complicated arguments.

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We now move away from the world of multiplicative prime number theory covered in Notes 1 and Notes 2, and enter the wider, and complementary, world of non-multiplicative prime number theory, in which one studies statistics related to non-multiplicative patterns, such as twins {n,n+2}. This creates a major jump in difficulty; for instance, even the most basic multiplicative result about the primes, namely Euclid’s theorem that there are infinitely many of them, remains unproven for twin primes. Of course, the situation is even worse for stronger results, such as Euler’s theorem, Dirichlet’s theorem, or the prime number theorem. Finally, even many multiplicative questions about the primes remain open. The most famous of these is the Riemann hypothesis, which gives the asymptotic {\sum_{n \leq x} \Lambda(n) = x + O( \sqrt{x} \log^2 x )} (see Proposition 24 from Notes 2). But even if one assumes the Riemann hypothesis, the precise distribution of the error term {O( \sqrt{x} \log^2 x )} in the above asymptotic (or in related asymptotics, such as for the sum {\sum_{x \leq n < x+y} \Lambda(n)} that measures the distribution of primes in short intervals) is not entirely clear.

Despite this, we do have a number of extremely convincing and well supported models for the primes (and related objects) that let us predict what the answer to many prime number theory questions (both multiplicative and non-multiplicative) should be, particularly in asymptotic regimes where one can work with aggregate statistics about the primes, rather than with a small number of individual primes. These models are based on taking some statistical distribution related to the primes (e.g. the primality properties of a randomly selected {k}-tuple), and replacing that distribution by a model distribution that is easy to compute with (e.g. a distribution with strong joint independence properties). One can then predict the asymptotic value of various (normalised) statistics about the primes by replacing the relevant statistical distributions of the primes with their simplified models. In this non-rigorous setting, many difficult conjectures on the primes reduce to relatively simple calculations; for instance, all four of the (still unsolved) Landau problems may now be justified in the affirmative by one or more of these models. Indeed, the models are so effective at this task that analytic number theory is in the curious position of being able to confidently predict the answer to a large proportion of the open problems in the subject, whilst not possessing a clear way forward to rigorously confirm these answers!

As it turns out, the models for primes that have turned out to be the most accurate in practice are random models, which involve (either explicitly or implicitly) one or more random variables. This is despite the prime numbers being obviously deterministic in nature; no coins are flipped or dice rolled to create the set of primes. The point is that while the primes have a lot of obvious multiplicative structure (for instance, the product of two primes is never another prime), they do not appear to exhibit much discernible non-multiplicative structure asymptotically, in the sense that they rarely exhibit statistical anomalies in the asymptotic limit that cannot be easily explained in terms of the multiplicative properties of the primes. As such, when considering non-multiplicative statistics of the primes, the primes appear to behave pseudorandomly, and can thus be modeled with reasonable accuracy by a random model. And even for multiplicative problems, which are in principle controlled by the zeroes of the Riemann zeta function, one can obtain good predictions by positing various pseudorandomness properties of these zeroes, so that the distribution of these zeroes can be modeled by a random model.

Of course, one cannot expect perfect accuracy when replicating a deterministic set such as the primes by a probabilistic model of that set, and each of the heuristic models we discuss below have some limitations to the range of statistics about the primes that they can expect to track with reasonable accuracy. For instance, many of the models about the primes do not fully take into account the multiplicative structure of primes, such as the connection with a zeta function with a meromorphic continuation to the entire complex plane; at the opposite extreme, we have the GUE hypothesis which appears to accurately model the zeta function, but does not capture such basic properties of the primes as the fact that the primes are all natural numbers. Nevertheless, each of the models described below, when deployed within their sphere of reasonable application, has (possibly after some fine-tuning) given predictions that are in remarkable agreement with numerical computation and with known rigorous theoretical results, as well as with other models in overlapping spheres of application; they are also broadly compatible with the general heuristic (discussed in this previous post) that in the absence of any exploitable structure, asymptotic statistics should default to the most “uniform”, “pseudorandom”, or “independent” distribution allowable.

As hinted at above, we do not have a single unified model for the prime numbers (other than the primes themselves, of course), but instead have an overlapping family of useful models that each appear to accurately describe some, but not all, aspects of the prime numbers. In this set of notes, we will discuss four such models:

  1. The Cramér random model and its refinements, which model the set {{\mathcal P}} of prime numbers by a random set.
  2. The Möbius pseudorandomness principle, which predicts that the Möbius function {\mu} does not correlate with any genuinely different arithmetic sequence of reasonable “complexity”.
  3. The equidistribution of residues principle, which predicts that the residue classes of a large number {n} modulo a small or medium-sized prime {p} behave as if they are independently and uniformly distributed as {p} varies.
  4. The GUE hypothesis, which asserts that the zeroes of the Riemann zeta function are distributed (at microscopic and mesoscopic scales) like the zeroes of a GUE random matrix, and which generalises the pair correlation conjecture regarding pairs of such zeroes.

This is not an exhaustive list of models for the primes and related objects; for instance, there is also the model in which the major arc contribution in the Hardy-Littlewood circle method is predicted to always dominate, and with regards to various finite groups of number-theoretic importance, such as the class groups discussed in Supplement 1, there are also heuristics of Cohen-Lenstra type. Historically, the first heuristic discussion of the primes along these lines was by Sylvester, who worked informally with a model somewhat related to the equidistribution of residues principle. However, we will not discuss any of these models here.

A word of warning: the discussion of the above four models will inevitably be largely informal, and “fuzzy” in nature. While one can certainly make precise formalisations of at least some aspects of these models, one should not be inflexibly wedded to a specific such formalisation as being “the” correct way to pin down the model rigorously. (To quote the statistician George Box: “all models are wrong, but some are useful”.) Indeed, we will see some examples below the fold in which some finer structure in the prime numbers leads to a correction term being added to a “naive” implementation of one of the above models to make it more accurate, and it is perfectly conceivable that some further such fine-tuning will be applied to one or more of these models in the future. These sorts of mathematical models are in some ways closer in nature to the scientific theories used to model the physical world, than they are to the axiomatic theories one is accustomed to in rigorous mathematics, and one should approach the discussion below accordingly. In particular, and in contrast to the other notes in this course, the material here is not directly used for proving further theorems, which is why we have marked it as “optional” material. Nevertheless, the heuristics and models here are still used indirectly for such purposes, for instance by

  • giving a clearer indication of what results one expects to be true, thus guiding one to fruitful conjectures;
  • providing a quick way to scan for possible errors in a mathematical claim (e.g. by finding that the main term is off from what a model predicts, or an error term is too small);
  • gauging the relative strength of various assertions (e.g. classifying some results as “unsurprising”, others as “potential breakthroughs” or “powerful new estimates”, others as “unexpected new phenomena”, and yet others as “way too good to be true”); or
  • setting up heuristic barriers (such as the parity barrier) that one has to resolve before resolving certain key problems (e.g. the twin prime conjecture).

See also my previous essay on the distinction between “rigorous” and “post-rigorous” mathematics, or Thurston’s essay discussing, among other things, the “definition-theorem-proof” model of mathematics and its limitations.

Remark 1 The material in this set of notes presumes some prior exposure to probability theory. See for instance this previous post for a quick review of the relevant concepts.

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In Notes 2, the Riemann zeta function {\zeta} (and more generally, the Dirichlet {L}-functions {L(\cdot,\chi)}) were extended meromorphically into the region {\{ s: \hbox{Re}(s) > 0 \}} in and to the right of the critical strip. This is a sufficient amount of meromorphic continuation for many applications in analytic number theory, such as establishing the prime number theorem and its variants. The zeroes of the zeta function in the critical strip {\{ s: 0 < \hbox{Re}(s) < 1 \}} are known as the non-trivial zeroes of {\zeta}, and thanks to the truncated explicit formulae developed in Notes 2, they control the asymptotic distribution of the primes (up to small errors).

The {\zeta} function obeys the trivial functional equation

\displaystyle  \zeta(\overline{s}) = \overline{\zeta(s)} \ \ \ \ \ (1)

for all {s} in its domain of definition. Indeed, as {\zeta(s)} is real-valued when {s} is real, the function {\zeta(s) - \overline{\zeta(\overline{s})}} vanishes on the real line and is also meromorphic, and hence vanishes everywhere. Similarly one has the functional equation

\displaystyle  \overline{L(s, \chi)} = L(\overline{s}, \overline{\chi}). \ \ \ \ \ (2)

From these equations we see that the zeroes of the zeta function are symmetric across the real axis, and the zeroes of {L(\cdot,\chi)} are the reflection of the zeroes of {L(\cdot,\overline{\chi})} across this axis.

It is a remarkable fact that these functions obey an additional, and more non-trivial, functional equation, this time establishing a symmetry across the critical line {\{ s: \hbox{Re}(s) = \frac{1}{2} \}} rather than the real axis. One consequence of this symmetry is that the zeta function and {L}-functions may be extended meromorphically to the entire complex plane. For the zeta function, the functional equation was discovered by Riemann, and reads as follows:

Theorem 1 (Functional equation for the Riemann zeta function) The Riemann zeta function {\zeta} extends meromorphically to the entire complex plane, with a simple pole at {s=1} and no other poles. Furthermore, one has the functional equation

\displaystyle  \zeta(s) = \alpha(s) \zeta(1-s) \ \ \ \ \ (3)

or equivalently

\displaystyle  \zeta(1-s) = \alpha(1-s) \zeta(s) \ \ \ \ \ (4)

for all complex {s} other than {s=0,1}, where {\alpha} is the function

\displaystyle  \alpha(s) := 2^s \pi^{s-1} \sin( \frac{\pi s}{2}) \Gamma(1-s). \ \ \ \ \ (5)

Here {\cos(z) := \frac{e^z + e^{-z}}{2}}, {\sin(z) := \frac{e^{-z}-e^{-z}}{2i}} are the complex-analytic extensions of the classical trigionometric functions {\cos(x), \sin(x)}, and {\Gamma} is the Gamma function, whose definition and properties we review below the fold.

The functional equation can be placed in a more symmetric form as follows:

Corollary 2 (Functional equation for the Riemann xi function) The Riemann xi function

\displaystyle  \xi(s) := \frac{1}{2} s(s-1) \pi^{-s/2} \Gamma(\frac{s}{2}) \zeta(s) \ \ \ \ \ (6)

is analytic on the entire complex plane {{\bf C}} (after removing all removable singularities), and obeys the functional equations

\displaystyle  \xi(\overline{s}) = \overline{\xi(s)}


\displaystyle  \xi(s) = \xi(1-s). \ \ \ \ \ (7)

In particular, the zeroes of {\xi} consist precisely of the non-trivial zeroes of {\zeta}, and are symmetric about both the real axis and the critical line. Also, {\xi} is real-valued on the critical line and on the real axis.

Corollary 2 is an easy consequence of Theorem 1 together with the duplication theorem for the Gamma function, and the fact that {\zeta} has no zeroes to the right of the critical strip, and is left as an exercise to the reader (Exercise 19). The functional equation in Theorem 1 has many proofs, but most of them are related in on way or another to the Poisson summation formula

\displaystyle  \sum_n f(n) = \sum_m \hat f(2\pi m) \ \ \ \ \ (8)

(Theorem 34 from Supplement 2, at least in the case when {f} is twice continuously differentiable and compactly supported), which can be viewed as a Fourier-analytic link between the coarse-scale distribution of the integers and the fine-scale distribution of the integers. Indeed, there is a quick heuristic proof of the functional equation that comes from formally applying the Poisson summation formula to the function {1_{x>0} \frac{1}{x^s}}, and noting that the functions {x \mapsto \frac{1}{x^s}} and {\xi \mapsto \frac{1}{\xi^{1-s}}} are formally Fourier transforms of each other, up to some Gamma function factors, as well as some trigonometric factors arising from the distinction between the real line and the half-line. Such a heuristic proof can indeed be made rigorous, and we do so below the fold, while also providing Riemann’s two classical proofs of the functional equation.

From the functional equation (and the poles of the Gamma function), one can see that {\zeta} has trivial zeroes at the negative even integers {-2,-4,-6,\dots}, in addition to the non-trivial zeroes in the critical strip. More generally, the following table summarises the zeroes and poles of the various special functions appearing in the functional equation, after they have been meromorphically extended to the entire complex plane, and with zeroes classified as “non-trivial” or “trivial” depending on whether they lie in the critical strip or not. (Exponential functions such as {2^{s-1}} or {\pi^{-s}} have no zeroes or poles, and will be ignored in this table; the zeroes and poles of rational functions such as {s(s-1)} are self-evident and will also not be displayed here.)

Function Non-trivial zeroes Trivial zeroes Poles
{\zeta(s)} Yes {-2,-4,-6,\dots} {1}
{\zeta(1-s)} Yes {1,3,5,\dots} {0}
{\sin(\pi s/2)} No Even integers No
{\cos(\pi s/2)} No Odd integers No
{\sin(\pi s)} No Integers No
{\Gamma(s)} No No {0,-1,-2,\dots}
{\Gamma(s/2)} No No {0,-2,-4,\dots}
{\Gamma(1-s)} No No {1,2,3,\dots}
{\Gamma((1-s)/2)} No No {2,4,6,\dots}
{\xi(s)} Yes No No

Among other things, this table indicates that the Gamma and trigonometric factors in the functional equation are tied to the trivial zeroes and poles of zeta, but have no direct bearing on the distribution of the non-trivial zeroes, which is the most important feature of the zeta function for the purposes of analytic number theory, beyond the fact that they are symmetric about the real axis and critical line. In particular, the Riemann hypothesis is not going to be resolved just from further analysis of the Gamma function!

The zeta function computes the “global” sum {\sum_n \frac{1}{n^s}}, with {n} ranging all the way from {1} to infinity. However, by some Fourier-analytic (or complex-analytic) manipulation, it is possible to use the zeta function to also control more “localised” sums, such as {\sum_n \frac{1}{n^s} \psi(\log n - \log N)} for some {N \gg 1} and some smooth compactly supported function {\psi: {\bf R} \rightarrow {\bf C}}. It turns out that the functional equation (3) for the zeta function localises to this context, giving an approximate functional equation which roughly speaking takes the form

\displaystyle  \sum_n \frac{1}{n^s} \psi( \log n - \log N ) \approx \alpha(s) \sum_m \frac{1}{m^{1-s}} \psi( \log M - \log m )

whenever {s=\sigma+it} and {NM = \frac{|t|}{2\pi}}; see Theorem 38 below for a precise formulation of this equation. Unsurprisingly, this form of the functional equation is also very closely related to the Poisson summation formula (8), indeed it is essentially a special case of that formula (or more precisely, of the van der Corput {B}-process). This useful identity relates long smoothed sums of {\frac{1}{n^s}} to short smoothed sums of {\frac{1}{m^{1-s}}} (or vice versa), and can thus be used to shorten exponential sums involving terms such as {\frac{1}{n^s}}, which is useful when obtaining some of the more advanced estimates on the Riemann zeta function.

We will give two other basic uses of the functional equation. The first is to get a good count (as opposed to merely an upper bound) on the density of zeroes in the critical strip, establishing the Riemann-von Mangoldt formula that the number {N(T)} of zeroes of imaginary part between {0} and {T} is {\frac{T}{2\pi} \log \frac{T}{2\pi} - \frac{T}{2\pi} + O(\log T)} for large {T}. The other is to obtain untruncated versions of the explicit formula from Notes 2, giving a remarkable exact formula for sums involving the von Mangoldt function in terms of zeroes of the Riemann zeta function. These results are not strictly necessary for most of the material in the rest of the course, but certainly help to clarify the nature of the Riemann zeta function and its relation to the primes.

In view of the material in previous notes, it should not be surprising that there are analogues of all of the above theory for Dirichlet {L}-functions {L(\cdot,\chi)}. We will restrict attention to primitive characters {\chi}, since the {L}-function for imprimitive characters merely differs from the {L}-function of the associated primitive factor by a finite Euler product; indeed, if {\chi = \chi' \chi_0} for some principal {\chi_0} whose modulus {q_0} is coprime to that of {\chi'}, then

\displaystyle  L(s,\chi) = L(s,\chi') \prod_{p|q_0} (1 - \frac{1}{p^s}) \ \ \ \ \ (9)

(cf. equation (45) of Notes 2).

The main new feature is that the Poisson summation formula needs to be “twisted” by a Dirichlet character {\chi}, and this boils down to the problem of understanding the finite (additive) Fourier transform of a Dirichlet character. This is achieved by the classical theory of Gauss sums, which we review below the fold. There is one new wrinkle; the value of {\chi(-1) \in \{-1,+1\}} plays a role in the functional equation. More precisely, we have

Theorem 3 (Functional equation for {L}-functions) Let {\chi} be a primitive character of modulus {q} with {q>1}. Then {L(s,\chi)} extends to an entire function on the complex plane, with

\displaystyle  L(s,\chi) = \varepsilon(\chi) 2^s \pi^{s-1} q^{1/2-s} \sin(\frac{\pi}{2}(s+\kappa)) \Gamma(1-s) L(1-s,\overline{\chi})

or equivalently

\displaystyle  L(1-s,\overline{\chi}) = \varepsilon(\overline{\chi}) 2^{1-s} \pi^{-s} q^{s-1/2} \sin(\frac{\pi}{2}(1-s+\kappa)) \Gamma(s) L(s,\chi)

for all {s}, where {\kappa} is equal to {0} in the even case {\chi(-1)=+1} and {1} in the odd case {\chi(-1)=-1}, and

\displaystyle  \varepsilon(\chi) := \frac{\tau(\chi)}{i^\kappa \sqrt{q}} \ \ \ \ \ (10)

where {\tau(\chi)} is the Gauss sum

\displaystyle  \tau(\chi) := \sum_{n \in {\bf Z}/q{\bf Z}} \chi(n) e(n/q). \ \ \ \ \ (11)

and {e(x) := e^{2\pi ix}}, with the convention that the {q}-periodic function {n \mapsto e(n/q)} is also (by abuse of notation) applied to {n} in the cyclic group {{\bf Z}/q{\bf Z}}.

From this functional equation and (2) we see that, as with the Riemann zeta function, the non-trivial zeroes of {L(s,\chi)} (defined as the zeroes within the critical strip {\{ s: 0 < \hbox{Re}(s) < 1 \}} are symmetric around the critical line (and, if {\chi} is real, are also symmetric around the real axis). In addition, {L(s,\chi)} acquires trivial zeroes at the negative even integers and at zero if {\chi(-1)=1}, and at the negative odd integers if {\chi(-1)=-1}. For imprimitive {\chi}, we see from (9) that {L(s,\chi)} also acquires some additional trivial zeroes on the left edge of the critical strip.

There is also a symmetric version of this equation, analogous to Corollary 2:

Corollary 4 Let {\chi,q,\varepsilon(\chi)} be as above, and set

\displaystyle  \xi(s,\chi) := (q/\pi)^{(s+\kappa)/2} \Gamma((s+\kappa)/2) L(s,\chi),

then {\xi(\cdot,\chi)} is entire with {\xi(1-s,\chi) = \varepsilon(\chi) \xi(s,\chi)}.

For further detail on the functional equation and its implications, I recommend the classic text of Titchmarsh or the text of Davenport.

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In Notes 1, we approached multiplicative number theory (the study of multiplicative functions {f: {\bf N} \rightarrow {\bf C}} and their relatives) via elementary methods, in which attention was primarily focused on obtaining asymptotic control on summatory functions {\sum_{n \leq x} f(n)} and logarithmic sums {\sum_{n \leq x} \frac{f(n)}{n}}. Now we turn to the complex approach to multiplicative number theory, in which the focus is instead on obtaining various types of control on the Dirichlet series {{\mathcal D} f}, defined (at least for {s} of sufficiently large real part) by the formula

\displaystyle  {\mathcal D} f(s) := \sum_n \frac{f(n)}{n^s}.

These series also made an appearance in the elementary approach to the subject, but only for real {s} that were larger than {1}. But now we will exploit the freedom to extend the variable {s} to the complex domain; this gives enough freedom (in principle, at least) to recover control of elementary sums such as {\sum_{n\leq x} f(n)} or {\sum_{n\leq x} \frac{f(n)}{n}} from control on the Dirichlet series. Crucially, for many key functions {f} of number-theoretic interest, the Dirichlet series {{\mathcal D} f} can be analytically (or at least meromorphically) continued to the left of the line {\{ s: \hbox{Re}(s) = 1 \}}. The zeroes and poles of the resulting meromorphic continuations of {{\mathcal D} f} (and of related functions) then turn out to control the asymptotic behaviour of the elementary sums of {f}; the more one knows about the former, the more one knows about the latter. In particular, knowledge of where the zeroes of the Riemann zeta function {\zeta} are located can give very precise information about the distribution of the primes, by means of a fundamental relationship known as the explicit formula. There are many ways of phrasing this explicit formula (both in exact and in approximate forms), but they are all trying to formalise an approximation to the von Mangoldt function {\Lambda} (and hence to the primes) of the form

\displaystyle  \Lambda(n) \approx 1 - \sum_\rho n^{\rho-1} \ \ \ \ \ (1)

where the sum is over zeroes {\rho} (counting multiplicity) of the Riemann zeta function {\zeta = {\mathcal D} 1} (with the sum often restricted so that {\rho} has large real part and bounded imaginary part), and the approximation is in a suitable weak sense, so that

\displaystyle  \sum_n \Lambda(n) g(n) \approx \int_0^\infty g(y)\ dy - \sum_\rho \int_0^\infty g(y) y^{\rho-1}\ dy \ \ \ \ \ (2)

for suitable “test functions” {g} (which in practice are restricted to be fairly smooth and slowly varying, with the precise amount of restriction dependent on the amount of truncation in the sum over zeroes one wishes to take). Among other things, such approximations can be used to rigorously establish the prime number theorem

\displaystyle  \sum_{n \leq x} \Lambda(n) = x + o(x) \ \ \ \ \ (3)

as {x \rightarrow \infty}, with the size of the error term {o(x)} closely tied to the location of the zeroes {\rho} of the Riemann zeta function.

The explicit formula (1) (or any of its more rigorous forms) is closely tied to the counterpart approximation

\displaystyle  -\frac{\zeta'}{\zeta}(s) \approx \frac{1}{s-1} - \sum_\rho \frac{1}{s-\rho} \ \ \ \ \ (4)

for the Dirichlet series {{\mathcal D} \Lambda = -\frac{\zeta'}{\zeta}} of the von Mangoldt function; note that (4) is formally the special case of (2) when {g(n) = n^{-s}}. Such approximations come from the general theory of local factorisations of meromorphic functions, as discussed in Supplement 2; the passage from (4) to (2) is accomplished by such tools as the residue theorem and the Fourier inversion formula, which were also covered in Supplement 2. The relative ease of uncovering the Fourier-like duality between primes and zeroes (sometimes referred to poetically as the “music of the primes”) is one of the major advantages of the complex-analytic approach to multiplicative number theory; this important duality tends to be rather obscured in the other approaches to the subject, although it can still in principle be discernible with sufficient effort.

More generally, one has an explicit formula

\displaystyle  \Lambda(n) \chi(n) \approx - \sum_\rho n^{\rho-1} \ \ \ \ \ (5)

for any (non-principal) Dirichlet character {\chi}, where {\rho} now ranges over the zeroes of the associated Dirichlet {L}-function {L(s,\chi) := {\mathcal D} \chi(s)}; we view this formula as a “twist” of (1) by the Dirichlet character {\chi}. The explicit formula (5), proven similarly (in any of its rigorous forms) to (1), is important in establishing the prime number theorem in arithmetic progressions, which asserts that

\displaystyle  \sum_{n \leq x: n = a\ (q)} \Lambda(n) = \frac{x}{\phi(q)} + o(x) \ \ \ \ \ (6)

as {x \rightarrow \infty}, whenever {a\ (q)} is a fixed primitive residue class. Again, the size of the error term {o(x)} here is closely tied to the location of the zeroes of the Dirichlet {L}-function, with particular importance given to whether there is a zero very close to {s=1} (such a zero is known as an exceptional zero or Siegel zero).

While any information on the behaviour of zeta functions or {L}-functions is in principle welcome for the purposes of analytic number theory, some regions of the complex plane are more important than others in this regard, due to the differing weights assigned to each zero in the explicit formula. Roughly speaking, in descending order of importance, the most crucial regions on which knowledge of these functions is useful are

  1. The region on or near the point {s=1}.
  2. The region on or near the right edge {\{ 1+it: t \in {\bf R} \}} of the critical strip {\{ s: 0 \leq \hbox{Re}(s) \leq 1 \}}.
  3. The right half {\{ s: \frac{1}{2} < \hbox{Re}(s) < 1 \}} of the critical strip.
  4. The region on or near the critical line {\{ \frac{1}{2} + it: t \in {\bf R} \}} that bisects the critical strip.
  5. Everywhere else.

For instance:

  1. We will shortly show that the Riemann zeta function {\zeta} has a simple pole at {s=1} with residue {1}, which is already sufficient to recover much of the classical theorems of Mertens discussed in the previous set of notes, as well as results on mean values of multiplicative functions such as the divisor function {\tau}. For Dirichlet {L}-functions, the behaviour is instead controlled by the quantity {L(1,\chi)} discussed in Notes 1, which is in turn closely tied to the existence and location of a Siegel zero.
  2. The zeta function is also known to have no zeroes on the right edge {\{1+it: t \in {\bf R}\}} of the critical strip, which is sufficient to prove (and is in fact equivalent to) the prime number theorem. Any enlargement of the zero-free region for {\zeta} into the critical strip leads to improved error terms in that theorem, with larger zero-free regions leading to stronger error estimates. Similarly for {L}-functions and the prime number theorem in arithmetic progressions.
  3. The (as yet unproven) Riemann hypothesis prohibits {\zeta} from having any zeroes within the right half {\{ s: \frac{1}{2} < \hbox{Re}(s) < 1 \}} of the critical strip, and gives very good control on the number of primes in intervals, even when the intervals are relatively short compared to the size of the entries. Even without assuming the Riemann hypothesis, zero density estimates in this region are available that give some partial control of this form. Similarly for {L}-functions, primes in short arithmetic progressions, and the generalised Riemann hypothesis.
  4. Assuming the Riemann hypothesis, further distributional information about the zeroes on the critical line (such as Montgomery’s pair correlation conjecture, or the more general GUE hypothesis) can give finer information about the error terms in the prime number theorem in short intervals, as well as other arithmetic information. Again, one has analogues for {L}-functions and primes in short arithmetic progressions.
  5. The functional equation of the zeta function describes the behaviour of {\zeta} to the left of the critical line, in terms of the behaviour to the right of the critical line. This is useful for building a “global” picture of the structure of the zeta function, and for improving a number of estimates about that function, but (in the absence of unproven conjectures such as the Riemann hypothesis or the pair correlation conjecture) it turns out that many of the basic analytic number theory results using the zeta function can be established without relying on this equation. Similarly for {L}-functions.

Remark 1 If one takes an “adelic” viewpoint, one can unite the Riemann zeta function {\zeta(\sigma+it) = \sum_n n^{-\sigma-it}} and all of the {L}-functions {L(\sigma+it,\chi) = \sum_n \chi(n) n^{-\sigma-it}} for various Dirichlet characters {\chi} into a single object, viewing {n \mapsto \chi(n) n^{-it}} as a general multiplicative character on the adeles; thus the imaginary coordinate {t} and the Dirichlet character {\chi} are really the Archimedean and non-Archimedean components respectively of a single adelic frequency parameter. This viewpoint was famously developed in Tate’s thesis, which among other things helps to clarify the nature of the functional equation, as discussed in this previous post. We will not pursue the adelic viewpoint further in these notes, but it does supply a “high-level” explanation for why so much of the theory of the Riemann zeta function extends to the Dirichlet {L}-functions. (The non-Archimedean character {\chi(n)} and the Archimedean character {n^{it}} behave similarly from an algebraic point of view, but not so much from an analytic point of view; as such, the adelic viewpoint is well suited for algebraic tasks (such as establishing the functional equation), but not for analytic tasks (such as establishing a zero-free region).)

Roughly speaking, the elementary multiplicative number theory from Notes 1 corresponds to the information one can extract from the complex-analytic method in region 1 of the above hierarchy, while the more advanced elementary number theory used to prove the prime number theorem (and which we will not cover in full detail in these notes) corresponds to what one can extract from regions 1 and 2.

As a consequence of this hierarchy of importance, information about the {\zeta} function away from the critical strip, such as Euler’s identity

\displaystyle  \zeta(2) = \frac{\pi^2}{6}

or equivalently

\displaystyle  1 + \frac{1}{2^2} + \frac{1}{3^2} + \dots = \frac{\pi^2}{6}

or the infamous identity

\displaystyle  \zeta(-1) = -\frac{1}{12},

which is often presented (slightly misleadingly, if one’s conventions for divergent summation are not made explicit) as

\displaystyle  1 + 2 + 3 + \dots = -\frac{1}{12},

are of relatively little direct importance in analytic prime number theory, although they are still of interest for some other, non-number-theoretic, applications. (The quantity {\zeta(2)} does play a minor role as a normalising factor in some asymptotics, see e.g. Exercise 28 from Notes 1, but its precise value is usually not of major importance.) In contrast, the value {L(1,\chi)} of an {L}-function at {s=1} turns out to be extremely important in analytic number theory, with many results in this subject relying ultimately on a non-trivial lower-bound on this quantity coming from Siegel’s theorem, discussed below the fold.

For a more in-depth treatment of the topics in this set of notes, see Davenport’s “Multiplicative number theory“.

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We will shortly turn to the complex-analytic approach to multiplicative number theory, which relies on the basic properties of complex analytic functions. In this supplement to the main notes, we quickly review the portions of complex analysis that we will be using in this course. We will not attempt a comprehensive review of this subject; for instance, we will completely neglect the conformal geometry or Riemann surface aspect of complex analysis, and we will also avoid using the various boundary convergence theorems for Taylor series or Dirichlet series (the latter type of result is traditionally utilised in multiplicative number theory, but I personally find them a little unintuitive to use, and will instead rely on a slightly different set of complex-analytic tools). We will also focus on the “local” structure of complex analytic functions, in particular adopting the philosophy that such functions behave locally like complex polynomials; the classical “global” theory of entire functions, while traditionally used in the theory of the Riemann zeta function, will be downplayed in these notes. On the other hand, we will play up the relationship between complex analysis and Fourier analysis, as we will incline to using the latter tool over the former in some of the subsequent material. (In the traditional approach to the subject, the Mellin transform is used in place of the Fourier transform, but we will not emphasise the role of the Mellin transform here.)

We begin by recalling the notion of a holomorphic function, which will later be shown to be essentially synonymous with that of a complex analytic function.

Definition 1 (Holomorphic function) Let {\Omega} be an open subset of {{\bf C}}, and let {f: \Omega \rightarrow {\bf C}} be a function. If {z \in {\bf C}}, we say that {f} is complex differentiable at {z} if the limit

\displaystyle  f'(z) := \lim_{h \rightarrow 0; h \in {\bf C} \backslash \{0\}} \frac{f(z+h)-f(z)}{h}

exists, in which case we refer to {f'(z)} as the (complex) derivative of {f} at {z}. If {f} is differentiable at every point {z} of {\Omega}, and the derivative {f': \Omega \rightarrow {\bf C}} is continuous, we say that {f} is holomorphic on {\Omega}.

Exercise 2 Show that a function {f: \Omega \rightarrow {\bf C}} is holomorphic if and only if the two-variable function {(x,y) \mapsto f(x+iy)} is continuously differentiable on {\{ (x,y) \in {\bf R}^2: x+iy \in \Omega\}} and obeys the Cauchy-Riemann equation

\displaystyle  \frac{\partial}{\partial x} f(x+iy) = \frac{1}{i} \frac{\partial}{\partial y} f(x+iy). \ \ \ \ \ (1)

Basic examples of holomorphic functions include complex polynomials

\displaystyle  P(z) = a_n z^n + \dots + a_1 z + a_0

as well as the complex exponential function

\displaystyle  \exp(z) := \sum_{n=0}^\infty \frac{z^n}{n!}

which are holomorphic on the entire complex plane {{\bf C}} (i.e., they are entire functions). The sum or product of two holomorphic functions is again holomorphic; the quotient of two holomorphic functions is holomorphic so long as the denominator is non-zero. Finally, the composition of two holomorphic functions is holomorphic wherever the composition is defined.

Exercise 3

  • (i) Establish Euler’s formula

    \displaystyle  \exp(x+iy) = e^x (\cos y + i \sin y)

    for all {x,y \in {\bf R}}. (Hint: it is a bit tricky to do this starting from the trigonometric definitions of sine and cosine; I recommend either using the Taylor series formulations of these functions instead, or alternatively relying on the ordinary differential equations obeyed by sine and cosine.)

  • (ii) Show that every non-zero complex number {z} has a complex logarithm {\log(z)} such that {\exp(\log(z))=z}, and that this logarithm is unique up to integer multiples of {2\pi i}.
  • (iii) Show that there exists a unique principal branch {\hbox{Log}(z)} of the complex logarithm in the region {{\bf C} \backslash (-\infty,0]}, defined by requiring {\hbox{Log}(z)} to be a logarithm of {z} with imaginary part between {-\pi} and {\pi}. Show that this principal branch is holomorphic with derivative {1/z}.

In real analysis, we have the fundamental theorem of calculus, which asserts that

\displaystyle  \int_a^b F'(t)\ dt = F(b) - F(a)

whenever {[a,b]} is a real interval and {F: [a,b] \rightarrow {\bf R}} is a continuously differentiable function. The complex analogue of this fact is that

\displaystyle  \int_\gamma F'(z)\ dz = F(\gamma(1)) - F(\gamma(0)) \ \ \ \ \ (2)

whenever {F: \Omega \rightarrow {\bf C}} is a holomorphic function, and {\gamma: [0,1] \rightarrow \Omega} is a contour in {\Omega}, by which we mean a piecewise continuously differentiable function, and the contour integral {\int_\gamma f(z)\ dz} for a continuous function {f} is defined via change of variables as

\displaystyle  \int_\gamma f(z)\ dz := \int_0^1 f(\gamma(t)) \gamma'(t)\ dt.

The complex fundamental theorem of calculus (2) follows easily from the real fundamental theorem and the chain rule.

In real analysis, we have the rather trivial fact that the integral of a continuous function on a closed contour is always zero:

\displaystyle  \int_a^b f(t)\ dt + \int_b^a f(t)\ dt = 0.

In complex analysis, the analogous fact is significantly more powerful, and is known as Cauchy’s theorem:

Theorem 4 (Cauchy’s theorem) Let {f: \Omega \rightarrow {\bf C}} be a holomorphic function in a simply connected open set {\Omega}, and let {\gamma: [0,1] \rightarrow \Omega} be a closed contour in {\Omega} (thus {\gamma(1)=\gamma(0)}). Then {\int_\gamma f(z)\ dz = 0}.

Exercise 5 Use Stokes’ theorem to give a proof of Cauchy’s theorem.

A useful reformulation of Cauchy’s theorem is that of contour shifting: if {f: \Omega \rightarrow {\bf C}} is a holomorphic function on a open set {\Omega}, and {\gamma, \tilde \gamma} are two contours in an open set {\Omega} with {\gamma(0)=\tilde \gamma(0)} and {\gamma(1) = \tilde \gamma(1)}, such that {\gamma} can be continuously deformed into {\tilde \gamma}, then {\int_\gamma f(z)\ dz = \int_{\tilde \gamma} f(z)\ dz}. A basic application of contour shifting is the Cauchy integral formula:

Theorem 6 (Cauchy integral formula) Let {f: \Omega \rightarrow {\bf C}} be a holomorphic function in a simply connected open set {\Omega}, and let {\gamma: [0,1] \rightarrow \Omega} be a closed contour which is simple (thus {\gamma} does not traverse any point more than once, with the exception of the endpoint {\gamma(0)=\gamma(1)} that is traversed twice), and which encloses a bounded region {U} in the anticlockwise direction. Then for any {z_0 \in U}, one has

\displaystyle  \int_\gamma \frac{f(z)}{z-z_0}\ dz= 2\pi i f(z_0).

Proof: Let {\varepsilon > 0} be a sufficiently small quantity. By contour shifting, one can replace the contour {\gamma} by the sum (concatenation) of three contours: a contour {\rho} from {\gamma(0)} to {z_0+\varepsilon}, a contour {C_\varepsilon} traversing the circle {\{z: |z-z_0|=\varepsilon\}} once anticlockwise, and the reversal {-\rho} of the contour {\rho} that goes from {z_0+\varepsilon} to {\gamma_0}. The contributions of the contours {\rho, -\rho} cancel each other, thus

\displaystyle \int_\gamma \frac{f(z)}{z-z_0}\ dz = \int_{C_\varepsilon} \frac{f(z)}{z-z_0}\ dz.

By a change of variables, the right-hand side can be expanded as

\displaystyle  2\pi i \int_0^1 f(z_0 + \varepsilon e^{2\pi i t})\ dt.

Sending {\varepsilon \rightarrow 0}, we obtain the claim. \Box

The Cauchy integral formula has many consequences. Specialising to the case when {\gamma} traverses a circle {\{ z: |z-z_0|=r\}} around {z_0}, we conclude the mean value property

\displaystyle  f(z_0) = \int_0^1 f(z_0 + re^{2\pi i t})\ dt \ \ \ \ \ (3)

whenever {f} is holomorphic in a neighbourhood of the disk {\{ z: |z-z_0| \leq r \}}. In a similar spirit, we have the maximum principle for holomorphic functions:

Lemma 7 (Maximum principle) Let {\Omega} be a simply connected open set, and let {\gamma} be a simple closed contour in {\Omega} enclosing a bounded region {U} anti-clockwise. Let {f: \Omega \rightarrow {\bf C}} be a holomorphic function. If we have the bound {|f(z)| \leq M} for all {z} on the contour {\gamma}, then we also have the bound {|f(z_0)| \leq M} for all {z_0 \in U}.

Proof: We use an argument of Landau. Fix {z_0 \in U}. From the Cauchy integral formula and the triangle inequality we have the bound

\displaystyle  |f(z_0)| \leq C_{z_0,\gamma} M

for some constant {C_{z_0,\gamma} > 0} depending on {z_0} and {\gamma}. This ostensibly looks like a weaker bound than what we want, but we can miraculously make the constant {C_{z_0,\gamma}} disappear by the “tensor power trick“. Namely, observe that if {f} is a holomorphic function bounded in magnitude by {M} on {\gamma}, and {n} is a natural number, then {f^n} is a holomorphic function bounded in magnitude by {M^n} on {\gamma}. Applying the preceding argument with {f, M} replaced by {f^n, M^n} we conclude that

\displaystyle  |f(z_0)|^n \leq C_{z_0,\gamma} M^n

and hence

\displaystyle  |f(z_0)| \leq C_{z_0,\gamma}^{1/n} M.

Sending {n \rightarrow \infty}, we obtain the claim. \Box

Another basic application of the integral formula is

Corollary 8 Every holomorphic function {f: \Omega \rightarrow {\bf C}} is complex analytic, thus it has a convergent Taylor series around every point {z_0} in the domain. In particular, holomorphic functions are smooth, and the derivative of a holomorphic function is again holomorphic.

Conversely, it is easy to see that complex analytic functions are holomorphic. Thus, the terms “complex analytic” and “holomorphic” are synonymous, at least when working on open domains. (On a non-open set {\Omega}, saying that {f} is analytic on {\Omega} is equivalent to asserting that {f} extends to a holomorphic function of an open neighbourhood of {\Omega}.) This is in marked contrast to real analysis, in which a function can be continuously differentiable, or even smooth, without being real analytic.

Proof: By translation, we may suppose that {z_0=0}. Let {C_r} be a a contour traversing the circle {\{ z: |z|=r\}} that is contained in the domain {\Omega}, then by the Cauchy integral formula one has

\displaystyle  f(z) = \frac{1}{2\pi i} \int_{C_r} \frac{f(w)}{w-z}\ dw

for all {z} in the disk {\{ z: |z| < r \}}. As {f} is continuously differentiable (and hence continuous) on {C_r}, it is bounded. From the geometric series formula

\displaystyle  \frac{1}{w-z} = \frac{1}{w} + \frac{1}{w^2} z + \frac{1}{w^3} z^2 + \dots

and dominated convergence, we conclude that

\displaystyle  f(z) = \sum_{n=0}^\infty (\frac{1}{2\pi i} \int_{C_r} \frac{f(w)}{w^{n+1}}\ dw) z^n

with the right-hand side an absolutely convergent series for {|z| < r}, and the claim follows. \Box

Exercise 9 Establish the generalised Cauchy integral formulae

\displaystyle  f^{(k)}(z_0) = \frac{k!}{2\pi i} \int_\gamma \frac{f(z)}{(z-z_0)^{k+1}}\ dz

for any non-negative integer {k}, where {f^{(k)}} is the {k}-fold complex derivative of {f}.

This in turn leads to a converse to Cauchy’s theorem, known as Morera’s theorem:

Corollary 10 (Morera’s theorem) Let {f: \Omega \rightarrow {\bf C}} be a continuous function on an open set {\Omega} with the property that {\int_\gamma f(z)\ dz = 0} for all closed contours {\gamma: [0,1] \rightarrow \Omega}. Then {f} is holomorphic.

Proof: We can of course assume {\Omega} to be non-empty and connected (hence path-connected). Fix a point {z_0 \in \Omega}, and define a “primitive” {F: \Omega \rightarrow {\bf C}} of {f} by defining {F(z_1) = \int_\gamma f(z)\ dz}, with {\gamma: [0,1] \rightarrow \Omega} being any contour from {z_0} to {z_1} (this is well defined by hypothesis). By mimicking the proof of the real fundamental theorem of calculus, we see that {F} is holomorphic with {F'=f}, and the claim now follows from Corollary 8. \Box

An important consequence of Morera’s theorem for us is

Corollary 11 (Locally uniform limit of holomorphic functions is holomorphic) Let {f_n: \Omega \rightarrow {\bf C}} be holomorphic functions on an open set {\Omega} which converge locally uniformly to a function {f: \Omega \rightarrow {\bf C}}. Then {f} is also holomorphic on {\Omega}.

Proof: By working locally we may assume that {\Omega} is a ball, and in particular simply connected. By Cauchy’s theorem, {\int_\gamma f_n(z)\ dz = 0} for all closed contours {\gamma} in {\Omega}. By local uniform convergence, this implies that {\int_\gamma f(z)\ dz = 0} for all such contours, and the claim then follows from Morera’s theorem. \Box

Now we study the zeroes of complex analytic functions. If a complex analytic function {f} vanishes at a point {z_0}, but is not identically zero in a neighbourhood of that point, then by Taylor expansion we see that {f} factors in a sufficiently small neighbourhood of {z_0} as

\displaystyle  f(z) = (z-z_0)^n g(z_0) \ \ \ \ \ (4)

for some natural number {n} (which we call the order or multiplicity of the zero at {f}) and some function {g} that is complex analytic and non-zero near {z_0}; this generalises the factor theorem for polynomials. In particular, the zero {z_0} is isolated if {f} does not vanish identically near {z_0}. We conclude that if {\Omega} is connected and {f} vanishes on a neighbourhood of some point {z_0} in {\Omega}, then it must vanish on all of {\Omega} (since the maximal connected neighbourhood of {z_0} in {\Omega} on which {f} vanishes cannot have any boundary point in {\Omega}). This implies unique continuation of analytic functions: if two complex analytic functions on {\Omega} agree on a non-empty open set, then they agree everywhere. In particular, if a complex analytic function does not vanish everywhere, then all of its zeroes are isolated, so in particular it has only finitely many zeroes on any given compact set.

Recall that a rational function is a function {f} which is a quotient {g/h} of two polynomials (at least outside of the set where {h} vanishes). Analogously, let us define a meromorphic function on an open set {\Omega} to be a function {f: \Omega \backslash S \rightarrow {\bf C}} defined outside of a discrete subset {S} of {\Omega} (the singularities of {f}), which is locally the quotient {g/h} of holomorphic functions, in the sense that for every {z_0 \in \Omega}, one has {f=g/h} in a neighbourhood of {z_0} excluding {S}, with {g, h} holomorphic near {z_0} and with {h} non-vanishing outside of {S}. If {z_0 \in S} and {g} has a zero of equal or higher order than {h} at {z_0}, then the singularity is removable and one can extend the meromorphic function holomorphically across {z_0} (by the holomorphic factor theorem (4)); otherwise, the singularity is non-removable and is known as a pole, whose order is equal to the difference between the order of {h} and the order of {g} at {z_0}. (If one wished, one could extend meromorphic functions to the poles by embedding {{\bf C}} in the Riemann sphere {{\bf C} \cup \{\infty\}} and mapping each pole to {\infty}, but we will not do so here. One could also consider non-meromorphic functions with essential singularities at various points, but we will have no need to analyse such singularities in this course.) If the order of a pole or zero is one, we say that it is simple; if it is two, we say it is double; and so forth.

Exercise 12 Show that the space of meromorphic functions on a non-empty open set {\Omega}, quotiented by almost everywhere equivalence, forms a field.

By quotienting two Taylor series, we see that if a meromorphic function {f} has a pole of order {n} at some point {z_0}, then it has a Laurent expansion

\displaystyle  f = \sum_{m=-n}^\infty a_m (z-z_0)^m,

absolutely convergent in a neighbourhood of {z_0} excluding {z_0} itself, and with {a_{-n}} non-zero. The Laurent coefficient {a_{-1}} has a special significance, and is called the residue of the meromorphic function {f} at {z_0}, which we will denote as {\hbox{Res}(f;z_0)}. The importance of this coefficient comes from the following significant generalisation of the Cauchy integral formula, known as the residue theorem:

Exercise 13 (Residue theorem) Let {f} be a meromorphic function on a simply connected domain {\Omega}, and let {\gamma} be a closed contour in {\Omega} enclosing a bounded region {U} anticlockwise, and avoiding all the singularities of {f}. Show that

\displaystyle  \int_\gamma f(z)\ dz = 2\pi i \sum_\rho \hbox{Res}(f;\rho)

where {\rho} is summed over all the poles of {f} that lie in {U}.

The residue theorem is particularly useful when applied to logarithmic derivatives {f'/f} of meromorphic functions {f}, because the residue is of a specific form:

Exercise 14 Let {f} be a meromorphic function on an open set {\Omega} that does not vanish identically. Show that the only poles of {f'/f} are simple poles (poles of order {1}), occurring at the poles and zeroes of {f} (after all removable singularities have been removed). Furthermore, the residue of {f'/f} at a pole {z_0} is an integer, equal to the order of zero of {f} if {f} has a zero at {z_0}, or equal to negative the order of pole at {f} if {f} has a pole at {z_0}.

Remark 15 The fact that residues of logarithmic derivatives of meromorphic functions are automatically integers is a remarkable feature of the complex analytic approach to multiplicative number theory, which is difficult (though not entirely impossible) to duplicate in other approaches to the subject. Here is a sample application of this integrality, which is challenging to reproduce by non-complex-analytic means: if {f} is meromorphic near {z_0}, and one has the bound {|\frac{f'}{f}(z_0+t)| \leq \frac{0.9}{t} + O(1)} as {t \rightarrow 0^+}, then {\frac{f'}{f}} must in fact stay bounded near {z_0}, because the only integer of magnitude less than {0.9} is zero.

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Analytic number theory is only one of many different approaches to number theory. Another important branch of the subject is algebraic number theory, which studies algebraic structures (e.g. groups, rings, and fields) of number-theoretic interest. With this perspective, the classical field of rationals {{\bf Q}}, and the classical ring of integers {{\bf Z}}, are placed inside the much larger field {\overline{{\bf Q}}} of algebraic numbers, and the much larger ring {{\mathcal A}} of algebraic integers, respectively. Recall that an algebraic number is a root of a polynomial with integer coefficients, and an algebraic integer is a root of a monic polynomial with integer coefficients; thus for instance {\sqrt{2}} is an algebraic integer (a root of {x^2-2}), while {\sqrt{2}/2} is merely an algebraic number (a root of {4x^2-2}). For the purposes of this post, we will adopt the concrete (but somewhat artificial) perspective of viewing algebraic numbers and integers as lying inside the complex numbers {{\bf C}}, thus {{\mathcal A} \subset \overline{{\bf Q}} \subset {\bf C}}. (From a modern algebraic perspective, it is better to think of {\overline{{\bf Q}}} as existing as an abstract field separate from {{\bf C}}, but which has a number of embeddings into {{\bf C}} (as well as into other fields, such as the completed p-adics {{\bf C}_p}), no one of which should be considered favoured over any other; cf. this mathOverflow post. But for the rudimentary algebraic number theory in this post, we will not need to work at this level of abstraction.) In particular, we identify the algebraic integer {\sqrt{-d}} with the complex number {\sqrt{d} i} for any natural number {d}.

Exercise 1 Show that the field of algebraic numbers {\overline{{\bf Q}}} is indeed a field, and that the ring of algebraic integers {{\mathcal A}} is indeed a ring, and is in fact an integral domain. Also, show that {{\bf Z} = {\mathcal A} \cap {\bf Q}}, that is to say the ordinary integers are precisely the algebraic integers that are also rational. Because of this, we will sometimes refer to elements of {{\bf Z}} as rational integers.

In practice, the field {\overline{{\bf Q}}} is too big to conveniently work with directly, having infinite dimension (as a vector space) over {{\bf Q}}. Thus, algebraic number theory generally restricts attention to intermediate fields {{\bf Q} \subset F \subset \overline{{\bf Q}}} between {{\bf Q}} and {\overline{{\bf Q}}}, which are of finite dimension over {{\bf Q}}; that is to say, finite degree extensions of {{\bf Q}}. Such fields are known as algebraic number fields, or number fields for short. Apart from {{\bf Q}} itself, the simplest examples of such number fields are the quadratic fields, which have dimension exactly two over {{\bf Q}}.

Exercise 2 Show that if {\alpha} is a rational number that is not a perfect square, then the field {{\bf Q}(\sqrt{\alpha})} generated by {{\bf Q}} and either of the square roots of {\alpha} is a quadratic field. Conversely, show that all quadratic fields arise in this fashion. (Hint: show that every element of a quadratic field is a root of a quadratic polynomial over the rationals.)

The ring of algebraic integers {{\mathcal A}} is similarly too large to conveniently work with directly, so in algebraic number theory one usually works with the rings {{\mathcal O}_F := {\mathcal A} \cap F} of algebraic integers inside a given number field {F}. One can (and does) study this situation in great generality, but for the purposes of this post we shall restrict attention to a simple but illustrative special case, namely the quadratic fields with a certain type of negative discriminant. (The positive discriminant case will be briefly discussed in Remark 42 below.)

Exercise 3 Let {d} be a square-free natural number with {d=1\ (4)} or {d=2\ (4)}. Show that the ring {{\mathcal O} = {\mathcal O}_{{\bf Q}(\sqrt{-d})}} of algebraic integers in {{\bf Q}(\sqrt{-d})} is given by

\displaystyle  {\mathcal O} = {\bf Z}[\sqrt{-d}] = \{ a + b \sqrt{-d}: a,b \in {\bf Z} \}.

If instead {d} is square-free with {d=3\ (4)}, show that the ring {{\mathcal O} = {\mathcal O}_{{\bf Q}(\sqrt{-d})}} is instead given by

\displaystyle  {\mathcal O} = {\bf Z}[\frac{1+\sqrt{-d}}{2}] = \{ a + b \frac{1+\sqrt{-d}}{2}: a,b \in {\bf Z} \}.

What happens if {d} is not square-free, or negative?

Remark 4 In the case {d=3\ (4)}, it may naively appear more natural to work with the ring {{\bf Z}[\sqrt{-d}]}, which is an index two subring of {{\mathcal O}}. However, because this ring only captures some of the algebraic integers in {{\bf Q}(\sqrt{-d})} rather than all of them, the algebraic properties of these rings are somewhat worse than those of {{\mathcal O}} (in particular, they generally fail to be Dedekind domains) and so are not convenient to work with in algebraic number theory.

We refer to fields of the form {{\bf Q}(\sqrt{-d})} for natural square-free numbers {d} as quadratic fields of negative discriminant, and similarly refer to {{\mathcal O}_{{\bf Q}(\sqrt{-d})}} as a ring of quadratic integers of negative discriminant. Quadratic fields and quadratic integers of positive discriminant are just as important to analytic number theory as their negative discriminant counterparts, but we will restrict attention to the latter here for simplicity of discussion.

Thus, for instance, when {d=1}, the ring of integers in {{\bf Q}(\sqrt{-1})} is the ring of Gaussian integers

\displaystyle  {\bf Z}[\sqrt{-1}] = \{ x + y \sqrt{-1}: x,y \in {\bf Z} \}

and when {d=3}, the ring of integers in {{\bf Q}(\sqrt{-3})} is the ring of Eisenstein integers

\displaystyle  {\bf Z}[\omega] := \{ x + y \omega: x,y \in {\bf Z} \}

where {\omega := e^{2\pi i /3}} is a cube root of unity.

As these examples illustrate, the additive structure of a ring {{\mathcal O} = {\mathcal O}_{{\bf Q}(\sqrt{-d})}} of quadratic integers is that of a two-dimensional lattice in {{\bf C}}, which is isomorphic as an additive group to {{\bf Z}^2}. Thus, from an additive viewpoint, one can view quadratic integers as “two-dimensional” analogues of rational integers. From a multiplicative viewpoint, however, the quadratic integers (and more generally, integers in a number field) behave very similarly to the rational integers (as opposed to being some sort of “higher-dimensional” version of such integers). Indeed, a large part of basic algebraic number theory is devoted to treating the multiplicative theory of integers in number fields in a unified fashion, that naturally generalises the classical multiplicative theory of the rational integers.

For instance, every rational integer {n \in {\bf Z}} has an absolute value {|n| \in {\bf N} \cup \{0\}}, with the multiplicativity property {|nm| = |n| |m|} for {n,m \in {\bf Z}}, and the positivity property {|n| > 0} for all {n \neq 0}. Among other things, the absolute value detects units: {|n| = 1} if and only if {n} is a unit in {{\bf Z}} (that is to say, it is multiplicatively invertible in {{\bf Z}}). Similarly, in any ring of quadratic integers {{\mathcal O} = {\mathcal O}_{{\bf Q}(\sqrt{-d})}} with negative discriminant, we can assign a norm {N(n) \in {\bf N} \cup \{0\}} to any quadratic integer {n \in {\mathcal O}_{{\bf Q}(\sqrt{-d})}} by the formula

\displaystyle  N(n) = n \overline{n}

where {\overline{n}} is the complex conjugate of {n}. (When working with other number fields than quadratic fields of negative discriminant, one instead defines {N(n)} to be the product of all the Galois conjugates of {n}.) Thus for instance, when {d=1,2\ (4)} one has

\displaystyle  N(x + y \sqrt{-d}) = x^2 + dy^2 \ \ \ \ \ (1)

and when {d=3\ (4)} one has

\displaystyle  N(x + y \frac{1+\sqrt{-d}}{2}) = x^2 + xy + \frac{d+1}{4} y^2. \ \ \ \ \ (2)

Analogously to the rational integers, we have the multiplicativity property {N(nm) = N(n) N(m)} for {n,m \in {\mathcal O}} and the positivity property {N(n) > 0} for {n \neq 0}, and the units in {{\mathcal O}} are precisely the elements of norm one.

Exercise 5 Establish the three claims of the previous paragraph. Conclude that the units (invertible elements) of {{\mathcal O}} consist of the four elements {\pm 1, \pm i} if {d=1}, the six elements {\pm 1, \pm \omega, \pm \omega^2} if {d=3}, and the two elements {\pm 1} if {d \neq 1,3}.

For the rational integers, we of course have the fundamental theorem of arithmetic, which asserts that every non-zero rational integer can be uniquely factored (up to permutation and units) as the product of irreducible integers, that is to say non-zero, non-unit integers that cannot be factored into the product of integers of strictly smaller norm. As it turns out, the same claim is true for a few additional rings of quadratic integers, such as the Gaussian integers and Eisenstein integers, but fails in general; for instance, in the ring {{\bf Z}[\sqrt{-5}]}, we have the famous counterexample

\displaystyle  6 = 2 \times 3 = (1+\sqrt{-5}) (1-\sqrt{-5})

that decomposes {6} non-uniquely into the product of irreducibles in {{\bf Z}[\sqrt{-5}]}. Nevertheless, it is an important fact that the fundamental theorem of arithmetic can be salvaged if one uses an “idealised” notion of a number in a ring of integers {{\mathcal O}}, now known in modern language as an ideal of that ring. For instance, in {{\bf Z}[\sqrt{-5}]}, the principal ideal {(6)} turns out to uniquely factor into the product of (non-principal) ideals {(2) + (1+\sqrt{-5}), (2) + (1-\sqrt{-5}), (3) + (1+\sqrt{-5}), (3) + (1-\sqrt{-5})}; see Exercise 27. We will review the basic theory of ideals in number fields (focusing primarily on quadratic fields of negative discriminant) below the fold.

The norm forms (1), (2) can be viewed as examples of positive definite quadratic forms {Q: {\bf Z}^2 \rightarrow {\bf Z}} over the integers, by which we mean a polynomial of the form

\displaystyle  Q(x,y) = ax^2 + bxy + cy^2

for some integer coefficients {a,b,c}. One can declare two quadratic forms {Q, Q': {\bf Z}^2 \rightarrow {\bf Z}} to be equivalent if one can transform one to the other by an invertible linear transformation {T: {\bf Z}^2 \rightarrow {\bf Z}^2}, so that {Q' = Q \circ T}. For example, the quadratic forms {(x,y) \mapsto x^2 + y^2} and {(x',y') \mapsto 2 (x')^2 + 2 x' y' + (y')^2} are equivalent, as can be seen by using the invertible linear transformation {(x,y) = (x',x'+y')}. Such equivalences correspond to the different choices of basis available when expressing a ring such as {{\mathcal O}} (or an ideal thereof) additively as a copy of {{\bf Z}^2}.

There is an important and classical invariant of a quadratic form {(x,y) \mapsto ax^2 + bxy + c y^2}, namely the discriminant {\Delta := b^2 - 4ac}, which will of course be familiar to most readers via the quadratic formula, which among other things tells us that a quadratic form will be positive definite precisely when its discriminant is negative. It is not difficult (particularly if one exploits the multiplicativity of the determinant of {2 \times 2} matrices) to show that two equivalent quadratic forms have the same discriminant. Thus for instance any quadratic form equivalent to (1) has discriminant {-4d}, while any quadratic form equivalent to (2) has discriminant {-d}. Thus we see that each ring {{\mathcal O}[\sqrt{-d}]} of quadratic integers is associated with a certain negative discriminant {D}, defined to equal {-4d} when {d=1,2\ (4)} and {-d} when {d=3\ (4)}.

Exercise 6 (Geometric interpretation of discriminant) Let {Q: {\bf Z}^2 \rightarrow {\bf Z}} be a quadratic form of negative discriminant {D}, and extend it to a real form {Q: {\bf R}^2 \rightarrow {\bf R}} in the obvious fashion. Show that for any {X>0}, the set {\{ (x,y) \in {\bf R}^2: Q(x,y) \leq X \}} is an ellipse of area {2\pi X / \sqrt{|D|}}.

It is natural to ask the converse question: if two quadratic forms have the same discriminant, are they necessarily equivalent? For certain choices of discriminant, this is the case:

Exercise 7 Show that any quadratic form {ax^2+bxy+cy^2} of discriminant {-4} is equivalent to the form {x^2+y^2}, and any quadratic form of discriminant {-3} is equivalent to {x^2+xy+y^2}. (Hint: use elementary transformations to try to make {|b|} as small as possible, to the point where one only has to check a finite number of cases; this argument is due to Legendre.) More generally, show that for any negative discriminant {D}, there are only finitely many quadratic forms of that discriminant up to equivalence (a result first established by Gauss).

Unfortunately, for most choices of discriminant, the converse question fails; for instance, the quadratic forms {x^2+5y^2} and {2x^2+2xy+3y^2} both have discriminant {-20}, but are not equivalent (Exercise 38). This particular failure of equivalence turns out to be intimately related to the failure of unique factorisation in the ring {{\bf Z}[\sqrt{-5}]}.

It turns out that there is a fundamental connection between quadratic fields, equivalence classes of quadratic forms of a given discriminant, and real Dirichlet characters, thus connecting the material discussed above with the last section of the previous set of notes. Here is a typical instance of this connection:

Proposition 8 Let {\chi_4: {\bf N} \rightarrow {\bf R}} be the real non-principal Dirichlet character of modulus {4}, or more explicitly {\chi_4(n)} is equal to {+1} when {n = 1\ (4)}, {-1} when {n = 3\ (4)}, and {0} when {n = 0,2\ (4)}.

  • (i) For any natural number {n}, the number of Gaussian integers {m \in {\bf Z}[\sqrt{-1}]} with norm {N(m)=n} is equal to {4(1 * \chi_4)(n)}. Equivalently, the number of solutions to the equation {n = x^2+y^2} with {x,y \in{\bf Z}} is {4(1*\chi_4)(n)}. (Here, as in the previous post, the symbol {*} denotes Dirichlet convolution.)
  • (ii) For any natural number {n}, the number of Gaussian integers {m \in {\bf Z}[\sqrt{-1}]} that divide {n} (thus {n = dm} for some {d \in {\bf Z}[\sqrt{-1}]}) is {4(1*1*1*\mu\chi_4)(n)}.

We will prove this proposition later in these notes. We observe that as a special case of part (i) of this proposition, we recover the Fermat two-square theorem: an odd prime {p} is expressible as the sum of two squares if and only if {p = 1\ (4)}. This proposition should also be compared with the fact, used crucially in the previous post to prove Dirichlet’s theorem, that {1*\chi(n)} is non-negative for any {n}, and at least one when {n} is a square, for any quadratic character {\chi}.

As an illustration of the relevance of such connections to analytic number theory, let us now explicitly compute {L(1,\chi_4)}.

Corollary 9 {L(1,\chi_4) = \frac{\pi}{4}}.

This particular identity is also known as the Leibniz formula.

Proof: For a large number {x}, consider the quantity

\displaystyle  \sum_{n \in {\bf Z}[\sqrt{-1}]: N(n) \leq x} 1

of all the Gaussian integers of norm less than {x}. On the one hand, this is the same as the number of lattice points of {{\bf Z}^2} in the disk {\{ (a,b) \in {\bf R}^2: a^2+b^2 \leq x \}} of radius {\sqrt{x}}. Placing a unit square centred at each such lattice point, we obtain a region which differs from the disk by a region contained in an annulus of area {O(\sqrt{x})}. As the area of the disk is {\pi x}, we conclude the Gauss bound

\displaystyle  \sum_{n \in {\bf Z}[\sqrt{-1}]: N(n) \leq x} 1 = \pi x + O(\sqrt{x}).

On the other hand, by Proposition 8(i) (and removing the {n=0} contribution), we see that

\displaystyle  \sum_{n \in {\bf Z}[\sqrt{-1}]: N(n) \leq x} 1 = 1 + 4 \sum_{n \leq x} 1 * \chi_4(n).

Now we use the Dirichlet hyperbola method to expand the right-hand side sum, first expressing

\displaystyle  \sum_{n \leq x} 1 * \chi_4(n) = \sum_{d \leq \sqrt{x}} \chi_4(d) \sum_{m \leq x/d} 1 + \sum_{m \leq \sqrt{x}} \sum_{d \leq x/m} \chi_4(d)

\displaystyle  - (\sum_{d \leq \sqrt{x}} \chi_4(d)) (\sum_{m \leq \sqrt{x}} 1)

and then using the bounds {\sum_{d \leq y} \chi_4(d) = O(1)}, {\sum_{m \leq y} 1 = y + O(1)}, {\sum_{d \leq \sqrt{x}} \frac{\chi_4(d)}{d} = L(1,\chi_4) + O(\frac{1}{\sqrt{x}})} from the previous set of notes to conclude that

\displaystyle  \sum_{n \leq x} 1 * \chi_4(n) = x L(1,\chi_4) + O(\sqrt{x}).

Comparing the two formulae for {\sum_{n \in {\bf Z}[\sqrt{-1}]: N(n) \leq x} 1} and sending {x \rightarrow \infty}, we obtain the claim. \Box

Exercise 10 Give an alternate proof of Corollary 9 that relies on obtaining asymptotics for the Dirichlet series {\sum_{n \in {\bf Z}} \frac{1 * \chi_4(n)}{n^s}} as {s \rightarrow 1^+}, rather than using the Dirichlet hyperbola method.

Exercise 11 Give a direct proof of Corollary 9 that does not use Proposition 8, instead using Taylor expansion of the complex logarithm {\log(1+z)}. (One can also use Taylor expansions of some other functions related to the complex logarithm here, such as the arctangent function.)

More generally, one can relate {L(1,\chi)} for a real Dirichlet character {\chi} with the number of inequivalent quadratic forms of a certain discriminant, via the famous class number formula; we will give a special case of this formula below the fold.

The material here is only a very rudimentary introduction to algebraic number theory, and is not essential to the rest of the course. A slightly expanded version of the material here, from the perspective of analytic number theory, may be found in Sections 5 and 6 of Davenport’s book. A more in-depth treatment of algebraic number theory may be found in a number of texts, e.g. Fröhlich and Taylor.

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In analytic number theory, an arithmetic function is simply a function {f: {\bf N} \rightarrow {\bf C}} from the natural numbers {{\bf N} = \{1,2,3,\dots\}} to the real or complex numbers. (One occasionally also considers arithmetic functions taking values in more general rings than {{\bf R}} or {{\bf C}}, as in this previous blog post, but we will restrict attention here to the classical situation of real or complex arithmetic functions.) Experience has shown that a particularly tractable and relevant class of arithmetic functions for analytic number theory are the multiplicative functions, which are arithmetic functions {f: {\bf N} \rightarrow {\bf C}} with the additional property that

\displaystyle f(nm) = f(n) f(m) \ \ \ \ \ (1)


whenever {n,m \in{\bf N}} are coprime. (One also considers arithmetic functions, such as the logarithm function {L(n) := \log n} or the von Mangoldt function, that are not genuinely multiplicative, but interact closely with multiplicative functions, and can be viewed as “derived” versions of multiplicative functions; see this previous post.) A typical example of a multiplicative function is the divisor function

\displaystyle \tau(n) := \sum_{d|n} 1 \ \ \ \ \ (2)


that counts the number of divisors of a natural number {n}. (The divisor function {n \mapsto \tau(n)} is also denoted {n \mapsto d(n)} in the literature.) The study of asymptotic behaviour of multiplicative functions (and their relatives) is known as multiplicative number theory, and is a basic cornerstone of modern analytic number theory.

There are various approaches to multiplicative number theory, each of which focuses on different asymptotic statistics of arithmetic functions {f}. In elementary multiplicative number theory, which is the focus of this set of notes, particular emphasis is given on the following two statistics of a given arithmetic function {f: {\bf N} \rightarrow {\bf C}}:

  1. The summatory functions

    \displaystyle \sum_{n \leq x} f(n)

    of an arithmetic function {f}, as well as the associated natural density

    \displaystyle \lim_{x \rightarrow \infty} \frac{1}{x} \sum_{n \leq x} f(n)

    (if it exists).

  2. The logarithmic sums

    \displaystyle \sum_{n\leq x} \frac{f(n)}{n}

    of an arithmetic function {f}, as well as the associated logarithmic density

    \displaystyle \lim_{x \rightarrow \infty} \frac{1}{\log x} \sum_{n \leq x} \frac{f(n)}{n}

    (if it exists).

Here, we are normalising the arithmetic function {f} being studied to be of roughly unit size up to logarithms, obeying bounds such as {f(n)=O(1)}, {f(n) = O(\log^{O(1)} n)}, or at worst

\displaystyle f(n) = O(n^{o(1)}). \ \ \ \ \ (3)


A classical case of interest is when {f} is an indicator function {f=1_A} of some set {A} of natural numbers, in which case we also refer to the natural or logarithmic density of {f} as the natural or logarithmic density of {A} respectively. However, in analytic number theory it is usually more convenient to replace such indicator functions with other related functions that have better multiplicative properties. For instance, the indicator function {1_{\mathcal P}} of the primes is often replaced with the von Mangoldt function {\Lambda}.

Typically, the logarithmic sums are relatively easy to control, but the summatory functions require more effort in order to obtain satisfactory estimates; see Exercise 7 below.

If an arithmetic function {f} is multiplicative (or closely related to a multiplicative function), then there is an important further statistic on an arithmetic function {f} beyond the summatory function and the logarithmic sum, namely the Dirichlet series

\displaystyle {\mathcal D}f(s) := \sum_{n=1}^\infty \frac{f(n)}{n^s} \ \ \ \ \ (4)


for various real or complex numbers {s}. Under the hypothesis (3), this series is absolutely convergent for real numbers {s>1}, or more generally for complex numbers {s} with {\hbox{Re}(s)>1}. As we will see below the fold, when {f} is multiplicative then the Dirichlet series enjoys an important Euler product factorisation which has many consequences for analytic number theory.

In the elementary approach to multiplicative number theory presented in this set of notes, we consider Dirichlet series only for real numbers {s>1} (and focusing particularly on the asymptotic behaviour as {s \rightarrow 1^+}); in later notes we will focus instead on the important complex-analytic approach to multiplicative number theory, in which the Dirichlet series (4) play a central role, and are defined not only for complex numbers with large real part, but are often extended analytically or meromorphically to the rest of the complex plane as well.

Remark 1 The elementary and complex-analytic approaches to multiplicative number theory are the two classical approaches to the subject. One could also consider a more “Fourier-analytic” approach, in which one studies convolution-type statistics such as

\displaystyle \sum_n \frac{f(n)}{n} G( t - \log n ) \ \ \ \ \ (5)


as {t \rightarrow \infty} for various cutoff functions {G: {\bf R} \rightarrow {\bf C}}, such as smooth, compactly supported functions. See for instance this previous blog post for an instance of such an approach. Another related approach is the “pretentious” approach to multiplicative number theory currently being developed by Granville-Soundararajan and their collaborators. We will occasionally make reference to these more modern approaches in these notes, but will primarily focus on the classical approaches.

To reverse the process and derive control on summatory functions or logarithmic sums starting from control of Dirichlet series is trickier, and usually requires one to allow {s} to be complex-valued rather than real-valued if one wants to obtain really accurate estimates; we will return to this point in subsequent notes. However, there is a cheap way to get upper bounds on such sums, known as Rankin’s trick, which we will discuss later in these notes.

The basic strategy of elementary multiplicative theory is to first gather useful estimates on the statistics of “smooth” or “non-oscillatory” functions, such as the constant function {n \mapsto 1}, the harmonic function {n \mapsto \frac{1}{n}}, or the logarithm function {n \mapsto \log n}; one also considers the statistics of periodic functions such as Dirichlet characters. These functions can be understood without any multiplicative number theory, using basic tools from real analysis such as the (quantitative version of the) integral test or summation by parts. Once one understands the statistics of these basic functions, one can then move on to statistics of more arithmetically interesting functions, such as the divisor function (2) or the von Mangoldt function {\Lambda} that we will discuss below. A key tool to relate these functions to each other is that of Dirichlet convolution, which is an operation that interacts well with summatory functions, logarithmic sums, and particularly well with Dirichlet series.

This is only an introduction to elementary multiplicative number theory techniques. More in-depth treatments may be found in this text of Montgomery-Vaughan, or this text of Bateman-Diamond.

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In the winter quarter (starting January 5) I will be teaching a graduate topics course entitled “An introduction to analytic prime number theory“. As the name suggests, this is a course covering many of the analytic number theory techniques used to study the distribution of the prime numbers {{\mathcal P} = \{2,3,5,7,11,\dots\}}. I will list the topics I intend to cover in this course below the fold. As with my previous courses, I will place lecture notes online on my blog in advance of the physical lectures.

The type of results about primes that one aspires to prove here is well captured by Landau’s classical list of problems:

  1. Even Goldbach conjecture: every even number {N} greater than two is expressible as the sum of two primes.
  2. Twin prime conjecture: there are infinitely many pairs {n,n+2} which are simultaneously prime.
  3. Legendre’s conjecture: for every natural number {N}, there is a prime between {N^2} and {(N+1)^2}.
  4. There are infinitely many primes of the form {n^2+1}.

All four of Landau’s problems remain open, but we have convincing heuristic evidence that they are all true, and in each of the four cases we have some highly non-trivial partial results, some of which will be covered in this course. We also now have some understanding of the barriers we are facing to fully resolving each of these problems, such as the parity problem; this will also be discussed in the course.

One of the main reasons that the prime numbers {{\mathcal P}} are so difficult to deal with rigorously is that they have very little usable algebraic or geometric structure that we know how to exploit; for instance, we do not have any useful prime generating functions. One of course can create non-useful functions of this form, such as the ordered parameterisation {n \mapsto p_n} that maps each natural number {n} to the {n^{th}} prime {p_n}, or one could invoke Matiyasevich’s theorem to produce a polynomial of many variables whose only positive values are prime, but these sorts of functions have no usable structure to exploit (for instance, they give no insight into any of the Landau problems listed above; see also Remark 2 below). The various primality tests in the literature, while useful for practical applications (e.g. cryptography) involving primes, have also proven to be of little utility for these sorts of problems; again, see Remark 2. In fact, in order to make plausible heuristic predictions about the primes, it is best to take almost the opposite point of view to the structured viewpoint, using as a starting point the belief that the primes exhibit strong pseudorandomness properties that are largely incompatible with the presence of rigid algebraic or geometric structure. We will discuss such heuristics later in this course.

It may be in the future that some usable structure to the primes (or related objects) will eventually be located (this is for instance one of the motivations in developing a rigorous theory of the “field with one element“, although this theory is far from being fully realised at present). For now, though, analytic and combinatorial methods have proven to be the most effective way forward, as they can often be used even in the near-complete absence of structure.

In this course, we will not discuss combinatorial approaches (such as the deployment of tools from additive combinatorics) in depth, but instead focus on the analytic methods. The basic principles of this approach can be summarised as follows:

  1. Rather than try to isolate individual primes {p} in {{\mathcal P}}, one works with the set of primes {{\mathcal P}} in aggregate, focusing in particular on asymptotic statistics of this set. For instance, rather than try to find a single pair {n,n+2} of twin primes, one can focus instead on the count {|\{ n \leq x: n,n+2 \in {\mathcal P} \}|} of twin primes up to some threshold {x}. Similarly, one can focus on counts such as {|\{ n \leq N: n, N-n \in {\mathcal P} \}|}, {|\{ p \in {\mathcal P}: N^2 < p < (N+1)^2 \}|}, or {|\{ n \leq x: n^2 + 1 \in {\mathcal P} \}|}, which are the natural counts associated to the other three Landau problems. In all four of Landau’s problems, the basic task is now to obtain a non-trivial lower bounds on these counts.
  2. If one wishes to proceed analytically rather than combinatorially, one should convert all these counts into sums, using the fundamental identity

    \displaystyle |A| = \sum_n 1_A(n),

    (or variants thereof) for the cardinality {|A|} of subsets {A} of the natural numbers {{\bf N}}, where {1_A} is the indicator function of {A} (and {n} ranges over {{\bf N}}). Thus we are now interested in estimating (and particularly in lower bounding) sums such as

    \displaystyle \sum_{n \leq N} 1_{{\mathcal P}}(n) 1_{{\mathcal P}}(N-n),

    \displaystyle \sum_{n \leq x} 1_{{\mathcal P}}(n) 1_{{\mathcal P}}(n+2),

    \displaystyle \sum_{N^2 < n < (N+1)^2} 1_{{\mathcal P}}(n),


    \displaystyle \sum_{n \leq x} 1_{{\mathcal P}}(n^2+1).

  3. Once one expresses number-theoretic problems in this fashion, we are naturally led to the more general question of how to accurately estimate (or, less ambitiously, to lower bound or upper bound) sums such as

    \displaystyle \sum_n f(n)

    or more generally bilinear or multilinear sums such as

    \displaystyle \sum_n \sum_m f(n,m)


    \displaystyle \sum_{n_1,\dots,n_k} f(n_1,\dots,n_k)

    for various functions {f} of arithmetic interest. (Importantly, one should also generalise to include integrals as well as sums, particularly contour integrals or integrals over the unit circle or real line, but we postpone discussion of these generalisations to later in the course.) Indeed, a huge portion of modern analytic number theory is devoted to precisely this sort of question. In many cases, we can predict an expected main term for such sums, and then the task is to control the error term between the true sum and its expected main term. It is often convenient to normalise the expected main term to be zero or negligible (e.g. by subtracting a suitable constant from {f}), so that one is now trying to show that a sum of signed real numbers (or perhaps complex numbers) is small. In other words, the question becomes one of rigorously establishing a significant amount of cancellation in one’s sums (also referred to as a gain or savings over a benchmark “trivial bound”). Or to phrase it negatively, the task is to rigorously prevent a conspiracy of non-cancellation, caused for instance by two factors in the summand {f(n)} exhibiting an unexpectedly large correlation with each other.

  4. It is often difficult to discern cancellation (or to prevent conspiracy) directly for a given sum (such as {\sum_n f(n)}) of interest. However, analytic number theory has developed a large number of techniques to relate one sum to another, and then the strategy is to keep transforming the sum into more and more analytically tractable expressions, until one arrives at a sum for which cancellation can be directly exhibited. (Note though that there is often a short-term tradeoff between analytic tractability and algebraic simplicity; in a typical analytic number theory argument, the sums will get expanded and decomposed into many quite messy-looking sub-sums, until at some point one applies some crude estimation to replace these messy sub-sums by tractable ones again.) There are many transformations available, ranging such basic tools as the triangle inequality, pointwise domination, or the Cauchy-Schwarz inequality to key identities such as multiplicative number theory identities (such as the Vaughan identity and the Heath-Brown identity), Fourier-analytic identities (e.g. Fourier inversion, Poisson summation, or more advanced trace formulae), or complex analytic identities (e.g. the residue theorem, Perron’s formula, or Jensen’s formula). The sheer range of transformations available can be intimidating at first; there is no shortage of transformations and identities in this subject, and if one applies them randomly then one will typically just transform a difficult sum into an even more difficult and intractable expression. However, one can make progress if one is guided by the strategy of isolating and enhancing a desired cancellation (or conspiracy) to the point where it can be easily established (or dispelled), or alternatively to reach the point where no deep cancellation is needed for the application at hand (or equivalently, that no deep conspiracy can disrupt the application).
  5. One particularly powerful technique (albeit one which, ironically, can be highly “ineffective” in a certain technical sense to be discussed later) is to use one potential conspiracy to defeat another, a technique I refer to as the “dueling conspiracies” method. This technique may be unable to prevent a single strong conspiracy, but it can sometimes be used to prevent two or more such conspiracies from occurring, which is particularly useful if conspiracies come in pairs (e.g. through complex conjugation symmetry, or a functional equation). A related (but more “effective”) strategy is to try to “disperse” a single conspiracy into several distinct conspiracies, which can then be used to defeat each other.

As stated before, the above strategy has not been able to establish any of the four Landau problems as stated. However, they can come close to such problems (and we now have some understanding as to why these problems remain out of reach of current methods). For instance, by using these techniques (and a lot of additional effort) one can obtain the following sample partial results in the Landau problems:

  1. Chen’s theorem: every sufficiently large even number {N} is expressible as the sum of a prime and an almost prime (the product of at most two primes). The proof proceeds by finding a nontrivial lower bound on {\sum_{n \leq N} 1_{\mathcal P}(n) 1_{{\mathcal E}_2}(N-n)}, where {{\mathcal E}_2} is the set of almost primes.
  2. Zhang’s theorem: There exist infinitely many pairs {p_n, p_{n+1}} of consecutive primes with {p_{n+1} - p_n \leq 7 \times 10^7}. The proof proceeds by giving a non-negative lower bound on the quantity {\sum_{x \leq n \leq 2x} (\sum_{i=1}^k 1_{\mathcal P}(n+h_i) - 1)} for large {x} and certain distinct integers {h_1,\dots,h_k} between {0} and {7 \times 10^7}. (The bound {7 \times 10^7} has since been lowered to {246}.)
  3. The Baker-Harman-Pintz theorem: for sufficiently large {x}, there is a prime between {x} and {x + x^{0.525}}. Proven by finding a nontrivial lower bound on {\sum_{x \leq n \leq x+x^{0.525}} 1_{\mathcal P}(n)}.
  4. The Friedlander-Iwaniec theorem: There are infinitely many primes of the form {n^2+m^4}. Proven by finding a nontrivial lower bound on {\sum_{n,m: n^2+m^4 \leq x} 1_{{\mathcal P}}(n^2+m^4)}.

We will discuss (simpler versions of) several of these results in this course.

Of course, for the above general strategy to have any chance of succeeding, one must at some point use some information about the set {{\mathcal P}} of primes. As stated previously, usefully structured parametric descriptions of {{\mathcal P}} do not appear to be available. However, we do have two other fundamental and useful ways to describe {{\mathcal P}}:

  1. (Sieve theory description) The primes {{\mathcal P}} consist of those numbers greater than one, that are not divisible by any smaller prime.
  2. (Multiplicative number theory description) The primes {{\mathcal P}} are the multiplicative generators of the natural numbers {{\bf N}}: every natural number is uniquely factorisable (up to permutation) into the product of primes (the fundamental theorem of arithmetic).

The sieve-theoretic description and its variants lead one to a good understanding of the almost primes, which turn out to be excellent tools for controlling the primes themselves, although there are known limitations as to how much information on the primes one can extract from sieve-theoretic methods alone, which we will discuss later in this course. The multiplicative number theory methods lead one (after some complex or Fourier analysis) to the Riemann zeta function (and other L-functions, particularly the Dirichlet L-functions), with the distribution of zeroes (and poles) of these functions playing a particularly decisive role in the multiplicative methods.

Many of our strongest results in analytic prime number theory are ultimately obtained by incorporating some combination of the above two fundamental descriptions of {{\mathcal P}} (or variants thereof) into the general strategy described above. In contrast, more advanced descriptions of {{\mathcal P}}, such as those coming from the various primality tests available, have (until now, at least) been surprisingly ineffective in practice for attacking problems such as Landau’s problems. One reason for this is that such tests generally involve operations such as exponentiation {a \mapsto a^n} or the factorial function {n \mapsto n!}, which grow too quickly to be amenable to the analytic techniques discussed above.

To give a simple illustration of these two basic approaches to the primes, let us first give two variants of the usual proof of Euclid’s theorem:

Theorem 1 (Euclid’s theorem) There are infinitely many primes.

Proof: (Multiplicative number theory proof) Suppose for contradiction that there were only finitely many primes {p_1,\dots,p_n}. Then, by the fundamental theorem of arithmetic, every natural number is expressible as the product of the primes {p_1,\dots,p_n}. But the natural number {p_1 \dots p_n + 1} is larger than one, but not divisible by any of the primes {p_1,\dots,p_n}, a contradiction.

(Sieve-theoretic proof) Suppose for contradiction that there were only finitely many primes {p_1,\dots,p_n}. Then, by the Chinese remainder theorem, the set of natural numbers {A} that is not divisible by any of the {p_1,\dots,p_n} has density {\prod_{i=1}^n (1-\frac{1}{p_i})}, that is to say

\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} | A \cap \{1,\dots,N\} | = \prod_{i=1}^n (1-\frac{1}{p_i}).

In particular, {A} has positive density and thus contains an element larger than {1}. But the least such element is one further prime in addition to {p_1,\dots,p_n}, a contradiction. \Box

Remark 1 One can also phrase the proof of Euclid’s theorem in a fashion that largely avoids the use of contradiction; see this previous blog post for more discussion.

Both proofs in fact extend to give a stronger result:

Theorem 2 (Euler’s theorem) The sum {\sum_{p \in {\mathcal P}} \frac{1}{p}} is divergent.

Proof: (Multiplicative number theory proof) By the fundamental theorem of arithmetic, every natural number is expressible uniquely as the product {p_1^{a_1} \dots p_n^{a_n}} of primes in increasing order. In particular, we have the identity

\displaystyle \sum_{n=1}^\infty \frac{1}{n} = \prod_{p \in {\mathcal P}} ( 1 + \frac{1}{p} + \frac{1}{p^2} + \dots )

(both sides make sense in {[0,+\infty]} as everything is unsigned). Since the left-hand side is divergent, the right-hand side is as well. But

\displaystyle ( 1 + \frac{1}{p} + \frac{1}{p^2} + \dots ) = \exp( \frac{1}{p} + O( \frac{1}{p^2} ) )

and {\sum_{p \in {\mathcal P}} \frac{1}{p^2}\leq \sum_{n=1}^\infty \frac{1}{n^2} < \infty}, so {\sum_{p \in {\mathcal P}} \frac{1}{p}} must be divergent.

(Sieve-theoretic proof) Suppose for contradiction that the sum {\sum_{p \in {\mathcal P}} \frac{1}{p}} is convergent. For each natural number {k}, let {A_k} be the set of natural numbers not divisible by the first {k} primes {p_1,\dots,p_k}, and let {A} be the set of numbers not divisible by any prime in {{\mathcal P}}. As in the previous proof, each {A_k} has density {\prod_{i=1}^k (1-\frac{1}{p_i})}. Also, since {\{1,\dots,N\}} contains at most {\frac{N}{p}} multiples of {p}, we have from the union bound that

\displaystyle | A \cap \{1,\dots,N \}| = |A_k \cap \{1,\dots,N\}| - O( N \sum_{i > k} \frac{1}{p_i} ).

Since {\sum_{i=1}^\infty \frac{1}{p_i}} is assumed to be convergent, we conclude that the density of {A_k} converges to the density of {A}; thus {A} has density {\prod_{i=1}^\infty (1-\frac{1}{p_i})}, which is non-zero by the hypothesis that {\sum_{i=1}^\infty \frac{1}{p_i}} converges. On the other hand, since the primes are the only numbers greater than one not divisible by smaller primes, {A} is just {\{1\}}, which has density zero, giving the desired contradiction. \Box

Remark 2 We have seen how easy it is to prove Euler’s theorem by analytic methods. In contrast, there does not seem to be any known proof of this theorem that proceeds by using any sort of prime-generating formula or a primality test, which is further evidence that such tools are not the most effective way to make progress on problems such as Landau’s problems. (But the weaker theorem of Euclid, Theorem 1, can sometimes be proven by such devices.)

The two proofs of Theorem 2 given above are essentially the same proof, as is hinted at by the geometric series identity

\displaystyle 1 + \frac{1}{p} + \frac{1}{p^2} + \dots = (1 - \frac{1}{p})^{-1}.

One can also see the Riemann zeta function begin to make an appearance in both proofs. Once one goes beyond Euler’s theorem, though, the sieve-theoretic and multiplicative methods begin to diverge significantly. On one hand, sieve theory can still handle to some extent sets such as twin primes, despite the lack of multiplicative structure (one simply has to sieve out two residue classes per prime, rather than one); on the other, multiplicative number theory can attain results such as the prime number theorem for which purely sieve theoretic techniques have not been able to establish. The deepest results in analytic number theory will typically require a combination of both sieve-theoretic methods and multiplicative methods in conjunction with the many transforms discussed earlier (and, in many cases, additional inputs from other fields of mathematics such as arithmetic geometry, ergodic theory, or additive combinatorics).

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Things are pretty quiet here during the holiday season, but one small thing I have been working on recently is a set of notes on special relativity that I will be working through in a few weeks with some bright high school students here at our local math circle.  I have only two hours to spend with this group, and it is unlikely that we will reach the end of the notes (in which I derive the famous mass-energy equivalence relation E=mc^2, largely following Einstein’s original derivation as discussed in this previous blog post); instead we will probably spend a fair chunk of time on related topics which do not actually require special relativity per se, such as spacetime diagrams, the Doppler shift effect, and an analysis of my airport puzzle.  This will be my first time doing something of this sort (in which I will be spending as much time interacting directly with the students as I would lecturing);  I’m not sure exactly how it will play out, being a little outside of my usual comfort zone of undergraduate and graduate teaching, but am looking forward to finding out how it goes.   (In particular, it may end up that the discussion deviates somewhat from my prepared notes.)

The material covered in my notes is certainly not new, but I ultimately decided that it was worth putting up here in case some readers here had any corrections or other feedback to contribute (which, as always, would be greatly appreciated).

[Dec 24 and then Jan 21: notes updated, in response to comments.]

I recently finished the first draft of the the first of my books, entitled “Hilbert’s fifth problem and related topics“, based on the lecture notes for my graduate course of the same name.    The PDF of this draft is available here.  As always, comments and corrections are welcome.


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